MP – 1
PARABOLA
C1
Conic Sections :
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed
point is in a constant ratio to its perpendicular distance from a fixed straight line and this constant ratio is
called eccentricity denoted by ‘e’.
Value of ‘e’
Types of conics
e=1
Parabola
0<e<1
Ellipse
e>1
Hyperbola
e=0
Circle
e=
Pair of straight lines
Important definitions related to conic section :

The fixed point is called the focus and any chord passing through the focus is called focal chord.

The fixed straight line is called the directrix and the line passing through the focus and perpendicular to the
directrix is called the axis.

A point of intersection of a conic with its axis is called a vertex.

A straight line drawn perpendicular to the axis and terminated at both end of the curve is a double ordinate
of the conic section.

The double ordinate passing through the focus is called the latus rectum of the conic section.

The point which bisects every chord of the conic passing through it, is called the centre of the conic
section.
C2
General equation of a conic :
The general equation of a conic with focus (, ) and directrix lx + my + n = 0 is :
(l2 + m2) [(x – p)2 + (y – q)2] = e2 (lx + my + n)2 which is equivalent to
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0. The determinant of the above equation is given by
 = abc + 2fgh – af2 – bg2 – ch2 = 0
C3
Recogination of different conics :
The nature of the conic section depends upon the position of the focus S w.r.t. the directrix and also upon
the value of the eccentricity e. Two different cases arise :
Case (I) when the focus lies on the directrix and  = 0 then the general equation of the conic represents a
pair of straight lines if
h2 > ab then the lines will be real and distinct intersecting at S.
h2 = ab then the lines will coincident.
h2 < ab then the lines will be imaginary
Case (II) when the focus does not lie on directrix and   0 then
a parabola
an ellipse
a hyperbola
rectangular hyperbola
e = 1;   0
0 < e < 1;   0;
e > 1;   0
e>1;0
2
2
h = ab
h < ab
2
h > ab
h2 > ab; a + b = 0
Practice Problems :
1.
What conic does the equation represent :
(i)
25[x2 + y2 – 2x + 1] = (4x – 3y + 1)2
(ii)
13x2 – 18xy + 37y2 + 2x + 14y – 2 = 0
(iii)
ax  by  1
[Answers : (1) (i) parabola (ii) ellipse (iii) parabola]
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 2
PARABOLA
C4
Definition and Some Terms related to Parabola
A parabola is the locus of a point, whose distance from a fixed point (focus) is equal to perpendicular
distance from a fixed straight line (directix). Four standard forms of the parabola are y2 = 4ax; y2 = –4ax;
x2 = 4ay; x2 = –4ay
For parabola y2 = 4ax
(i)
Vertex is A(0, 0)
(ii)
focus is S(a, 0)
(iii)
Axis is y = 0
(iv)
Directrix is x + a = 0
(v)
Focal Distance : The distance of a point on the parabola from the focus i.e., SP
(vi)
Focal Chord : A chord of the parabola, which passes through the focus i.e., SP
(vii)
Double Ordinate : A chord of the parabola perpendicular to the axis of the symmetry i.e., PP
Latus Rectum : A double ordinate passing through the focus or a focal chord perpendicular to
the axis of parabola is called the Latus Rectum (L.R.) i.e., LL
(viii)
For y2 = 4ax, the ends of the latus rectum are L(a, 2a) and L’(a, –2a) and length of the latus
rectum = 4a
(ix)
Parametric Representation : The simplest and the best form of representing the co-ordinates
of a point on the parabola is (at2, 2at) i.e. the equation x = at2 and y = 2at together represents the
parabola y2 = 4ax, t being the parameter.
Some other Points :
(i)
Perpendicular distance from focus on directrix = half the latus rectum.
(ii)
Vertex is middle point of the focus and the point of intersection of directrix and axis.
(iii)
Two parabolas are said to be equal if they have the same latus rectum.
(iv)
Other forms of parabola
(1)
Einstein Classes,
Parabola y2 = –4ax, a > 0
(a)
Vertex is A(0, 0)
(b)
Focus is S(–a, 0)
(c)
Equation of the directrix MZ is x – a = 0
(d)
Equation of the axis is y = 0 i.e., x-axis
(e)
Equation of the tangent at the vertex is x = 0 i.e., y-axis
(f)
Length of latus rectum = LL   4a
(g)
Extremities of latus rectum are L (–a, 2a) and L (–a, –2a)
(h)
Equation of latus rectum is x = – a i.e., x + a = 0
(i)
Parametric co-ordinates is (–at2, 2at)
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 3
(2)
(3)
Parabola x2 = 4ay, a > 0
(a)
Vertex is A(0, 0)
(b)
Focus is S (0, a)
(c)
Equation of the directrix MZ is y + a = 0
(d)
Equation of the axis is x = 0 i.e., y-axis
(e)
Equation of the tangent at the vertex is y = 0 i.e.,
x-axis
(f)
Extremities of latus rectum are L(2a, a)
and L( 2a, a) .
(g)
Length of latus rectum = LL   4a
(h)
Equation of latus rectum is y = a i.e., y – a = 0
(i)
Parametric co-ordinates is (2at, at2)
Parabola x2 = –4ay, a > 0
(a)
Vertex is A(0, 0)
(b)
Focus is S (0, –a)
(c)
Equation of the directrix MZ is y – a = 0
(d)
Equation of the axis is x = 0 i.e., y-axis
(e)
Equation of the tangent at the vertex is y = 0
i.e., x-axis
(v)
(f)
Length of latus rectum = LL   4a
(g)
Extremities of latus rectum are L(2a, –a) and L(2a,  a)
(h)
Equation of latus rectum is y = – a i.e., y + a = 0
(i)
Parametric co-ordinates is (2at2, – at2)
Position of a point Relative to a Parabola : The point (x1, y1) lies outside, on or inside the
parabola y2 = 4ax according as the expression y12 – 4ax1 is positive, zero or negative.
Practice Problems :
1.
2.
3.
Find the equation of the parabola
(i)
Focus is at (–1, –2) and the directrix is the straight line x – 2y + 3 = 0
(ii)
Focus is at (4, –3) and vertex is (4, –1)
PP is a double ordinate of a parabola y2 = 4ax. Find the locus of its point of trisection.
Determine the name of the curve described parametrically by the equations
x = t2 + t + 1, y = t2 – t + 1.
4.
What is the position of the point (2, 3) with respect to the parabola y2 = 3x.
5.
Find the vertex, focus, directrix and length of the latus rectum of the parabola y2 – 4y – 2x – 8 = 0.
6.
Find the equation of the parabola the extremities of whose latus rectum are (1, 2) and (1, –4).
[Answers : (1) (i) 4x2 + y2 + 4xy + 4x + 32y + 16 = 0 (ii) x2 – 8x + 8y + 24 = 0 (2) 9y2 = 4ax (3) Parabola
(4) outside]
C5
Line and a Parabola :
The line y = mx + c meets the parabola y2 = 4ax in two points real if a > mc, coincident if a = mc or
imaginary if a < mc. The condition of tangency is c = a/m.
 4 
Length of the chord intercepted by the parabola on the line y = mx + c is  2  a(1  m 2 )(a  mc )
m 
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 4
In Parametric Form :
1.
The equation of a chord joining t1 and t2 is 2x – (t1 + t2)y + 2at1t2 = 0
2.
If t1 and t2 are the ends of a focal chord of the parabola y2 = 4ax then t1t2 = –1. Hence the
2a 
a
, 
2
t 
t
co-ordinates at the extremities of a focal chord can be taken as (at2, 2at) and 
Practice Problems :
1.
If the point (at2, 2at) be the extremity of a focal chord of parabola y2 = 4ax then show that the length
2
 1
of the focal chord is a t   .
t

2.
Prove that the semi latus rectum of the parabola y2 = 4ax is the harmonic mean between the segment
of any focal chord of the parabola.
3.
Show that the focal chord of parabola y2 = 4ax makes an angle  with the x-axis is of length
4a cosec2.
4.
Prove that the straight line lx + my + n = 0 touches the parabola y2 = 4ax if ln = am2.
5.
Show that the line x cos  + y sin  = p touches the parabola y2 = 4ax if p cos  + a sin2 = 0 and that
the point of contact is (a tan2, –2a tan ).
C6
Equation of Tangents to the Parabola y2 = 4ax :
(i)
at the point (x1, y1) is yy1 = 2a(x + x1)
(ii)
a
 a 2a 
at the point  2 ,  is y  mx  (m  0) at
m
m m 
(iii)
at the point (at2, 2at) or t is ty = x + at2
Point of intersection of the tangents at the point t1 and t2 is [at1t2, a(t1 + t2)]
Practice Problems :
1.
Find the equation of the common tangents to the parabola y2 = 4ax and x2 = 4by.
2.
The tangent to the parabola y2 = 4ax make angle 1 and 2 with x-axis. Find the locus of the their
point of intersection if cot 1 + cot 2 = c.
3.
The tangent to the parabola y2 = 4ax at P(at21, 2at1) and Q(at22, 2at2) intersect at R. Prove that the
area of the triangle PQR is
1 2
a | (t 1  t 2 ) |3 .
2
4.
Find the locus of the point of intersection of the tangents to the parabola y2 = 4ax which include an
angle .
5.
Find the common tangents of x2 + y2 = 2a2 and y2 = 8ax
[Answers : (1) a1/3x + b1/3y + a2/3b2/3 = 0 (2) y = ac]
C7
Equation of Normals to the parabola y2 = 4ax :
(i)
at (x1, y1) is y  y 1  
y1
( x  x1 ) ;
2a
(ii)
at (am2, –2am) is y = mx – 2am – am3
(iii)
at (at2, 2at) is y + tx = 2at + at3
Important Points :
(i)
Point of intersection of normals at t1 and t2 are a [(t12 + t22 + t1t2 + 2), –at1t2(t1 + t2)].
(ii)
If the normals to the parabola y2 = 4ax at the point t1 meets the parabola again at the point t2

2
then t2 =   t 1   .
t1 

Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 5
(iii)
If the normals to the parabola y2 = 4ax at the points t1 and t2 intersect again on the parabola at
the point ‘t3’ then t1t2 = 2; t3 = –(t1 + t2) and the line joining t1 and t2 passes through a fixed point
(–2a, 0).
Practice Problems :
1.
Show that normal to the parabola y2 = 8x at the point (2, 4) meets it again at (18, –12). Find also the
length of the normal chord.
2.
Prove that the chord y – x2 + 4a2 = 0 is a normal chord of the parabola y2 = 4ax. Also, find the
point on the parabola when the given chord is normal to the parabola.
3.
Prove that the normal chord to a parabola y2 = 4ax at the point whose ordinate is equal to abscissa
subtends a right angle at the focus.
4.
The ordinates of points P and Q on the parabola y2 = 12x are in the ratio 1 : 2. Find the locus of the
point of intersection of the normals to the parabola at P and Q.
[Answers : (1) 162 (2) (2a, –22a]
C8
Co-normal points :
In general three normals can be drawn from a point to a parabola and the points where they meet the
parabola are called co-normal points.
Practice Problems :
1.
Consider a point (h, k) from which three normals are drawn to the parabola y2 = 4ax. If the slopes of
these normals are m1, m2 and m3 then prove that (i) m1 + m2 + m3 = 0 (ii) m1m2 + m2m3 + m3m1 =
( 2a  h )
k
(iii) m1m2m3 =  (iv) The algebraic sum of ordinates of the feets of three normals drawn
a
a
to a parabola from a given point is zero (v) If three normals be real then h > 2a (vi) The centroid of
the triangle formed by the feet of the three normals lies on the axis of the parabola (vii) If three
normals be real and distinct then 27ak2 < 4(h – 2a)3.
2.
Show that the locus of points such that two of the three normals drawn from them to the parabola
y2 = 4ax coincide is 27ay2 = 4(x – 2a)3.
3.
Find the locus of the point through which pass three normals to the parabola y2 = 4ax such that
two of them make angles  and  respectively with the axis such that tan  tan  = 2.
4.
Consider a point (h, k) from which three normals are drawn to the parabola y2 = 4ax. Find the
equation of the circle which will pass through the three co-normal points of this parabola.
[Answers : (3) y2 – 4ax = 0 (4) x 2  y 2  ( 2a  h )x 
C9
k
y  0]
2
Pair of Tangents :
The equation to the pair of tangents which can be drawn from any outside point (x1, y1) to the parabola
y2 = 4ax is given by SS1 = T2 where S  y2 – 4ax, S1 = y12 – 4ax1, T  y y1 – 2a(x + x1)
C10
Director Cicle :
Locus of the point of intersection of the perpendicular tangents to a curve is called the Director Circle. For
parabola y2 = 4ax it’s equation is x + a = 0 which is parabola’s own directrix.
C11
Chord of Contact :
The chord joining the points of contact of two tangents drawn from an external point to a parabola is known
as the chord of contact of tangents drawn from external point. Equation to the chord of contact of tangents
drawn from a point P(x1, y1) to the parabola y2 = 4ax is yy1 = 2a (x + x1).
C12
Chord with a given middle point :
Equation of the chord of the parabola y2 = 4ax whose middle point is (x1, y1) is y – y1 =
2a
( x  x1 ) or T = S1
y1
where T = yy1 – 2a(x + x1) and S1 = y12 – 4ax1
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 6
Practice Problems :
1.
Prove that the area of the triangle formed by the tangents from the point (x1, y1) and the chord of
contact is (y12 – 4ax1)3/2 / 2a.
2.
Tangents are drawn from the point (x1, y1) to the parabola y2 = 4ax, show that the length of their
1
( y 12  4ax1 )( y 12  4a 2 ) .
|a|
chord of contact is
3.
Find the locus of the mid points of the chords of the parabola y2 = 4ax which subtend a right angle at
the vertex of the parabola.
4.
Show that the locus of the middle points of normal chords of the parabola y2 = 4ax is
y4 – 2a(x – 2a)y2 + 8a4 = 0.
5.
Through the vertex O of a parabola y2 = 4x chords OP and OQ are drawn at right angles to one
another. Show that for all positions of P, PQ cuts the axis of the parabola at a fixed point. Also find
the locus of the middle point of PQ.
[Answers : (3) y2 – 2ax + 8a2 = 0]
C13
Diameter :
The locus of the middle points of a system of parallel chords is called a diameter and in case of a parabola
this diameter is shown to be a straight line which is parallel to the axis of the parabola.
Practice Problems :
1.
Find the equation of the diameter bisecting chords of slope m of the parabola y2 = 4ax ?
2.
If the diameter through any point P of a parabola meets any chord in A and the tangent at the end of
the chord meets the diameter in B and C, then prove that PA2 = PB · PC.
[Answers : (1) y 
C14
2a
]
m
Pole and Polar
From any point P (inside or outside the parabola), a straight line is drawn which intersects the parabola at
Q and R. Now the tangents are drawn to the parabola at Q and R. The locus of the point of intersection of
these tangents is called the polar of the given point P with respect to parabola and the point P is called the
pole of the polar.
Practice Problems :
1.
From any point P(x1, y1) (inside or outside the parabola), a straight line is drawn which intersects the
parabola y2 = 4ax at Q and R. Now the tangents are drawn to the parabola at Q and R. Find the locus
of the point of intersection of these tangents ?
2.
Find the pole of the given line px + qy + r = 0 with respect to the parabola y2 = 4ax ?
3.
Find the locus of the poles of normal chord of y2 = 4ax.
4.
Find the locus of the poles of the chords of the parabola y2 = 4ax which subtend a constant angle  at
the vertex.
 r  2aq 
 (3) (x + 2a)y2 + 4a3 = 0 (4) (4a + x)2tan2 = 4(y2 – 4ax)]
[Answers : (1) yy1 = 2a(x + x1) (2)  ,
p 
p
C15
Reflection theory of a Parabolic Mirror :
Any set of rays, parallel to the axis of the parabola, are incident on a parabolic reflector then after the
reflection, these rays pass through the focus of the parabola.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 7
Practice Problems :
1.
A ray of light moving parallel to x-axis given by y = b is incident on a parabolic mirror represented
by the equation y2 = 4ax. Find the equation of the reflected ray.
2.
The focus of a parabolic mirror as shown in figure is at a distance of 5 cm from its vertex. If the
mirror is 45 cm deep. Then find the distance AB ?
[Answers : (1) (y – b) (4a2 – b2) = –(4ax – b2) (2) 60 cm]
C16
Important Highlights :
(i)
If the tangent and normal at any point ‘P’ of the parabola intersect the axis at T and G then
ST = SG = SP where ‘S’ is the focus. In other words the tangent and the normal at a point P on the
parabola are the bisector of the angle between the focal radius SP and the perpendicular from P
on the directrix. From this we conclude that all rays emanating from S will become parallel to the
axis of the parabola after reflection
(ii)
The portion of a tangent to a parabola cut off between the directrix and the curve subtends a right
angle at the focus.
(iii)
The tangent at the extremities of a focal chord intersect at right angles on the directrix, and hence
a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii of a
point P (at2, 2at) as diameter touches the tangent at the vertex and intercepts a chord of length
a 1  t 2 on a normal at the point P..
(iv)
Any tangent to a parabola and the perpendicular on it from the focus meet on the tangent at the
vertex.
(v)
Semi latus rectum of the parabola y2 = 4ax, is the harmonic mean between segments of any focal
chord of the parabola.
(vi)
The area of the triangle formed by three points on a parabola is twice the area of the triangle
formed by the tangents at these points.
(vii)
Length of subtangent at any point P(x, y) on the parabola y2 = 4ax equals twice the abscissa
of the point P. Note that the subtangent is bisected at the vertex.
(viii)
Length of subnormal is constant for all points on the parabola and is equal to the semi latus
rectum.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 8
SINGLE CORRECT CHOICE TYPE
1.
2.
3.
4.
5.
6.
(a)
a pair of straight lines
It (at2, 2at) are the co-ordinates of one end of a
focal chord of the parabola y 2 = 4ax, then the
co-ordinates of the other end are
(b)
an ellipse
(a)
(at2, – 2at)
(b)
(–at2, –2at)
(c)
a parabola
(d)
a hyperbola
(c)
 a 2a 
 2, 
t t 
(d)
 a 2a 
 2 , 
t 
t
The curve described parametrically
x = t2 + t + 1, y = t2 – t + 1 represents :
by
Consider a circle with its centre lying on the focus
of the parabola y2 = 2p x such that it touches the
directrix of the parabola. Then, a point of
intersection of the circle and the parabola is
(a)
p 
 , p
2 
(b)
p

 ,p 
2

(c)
 p 
 , p
 2 
(d)
 p

 ,  p
 2

7.
8.
9.
Three normals to the parabola y2 = x drawn through
a point (c, 0), then
(a)
c
1
4
(b)
(c)
c
1
2
(d)
1
2
none of these
If the normals from any point to the parabola
x2 = 4y cuts the line y = 2 in points whose abscissae
are in A.P., then the slopes of the tangents at 3
co-normal points are in
(a)
A.P.
(b)
(c)
H.P.
(d)
10.
11.
G.P.
none of these
The length of the chord of the parabola x2 = 4ay
passing through the vertex and having slope tan 
is
12.
If P S Q is the focal chord of the parabola
y2 = 8 x such that S P = 6. Then length S Q is
(a)
6
(c)
3
(b)
(d)
The point on the curve y = ax, the tangent at which
makes an angle of 450 with x-axis will be given by
(a)
a a
 , 
2 4
(b)
 a a
 , 
 2 4
(c)
a a
 , 
4 2
(d)
 a a
 , 
 4 2
The number of point (s) (x, y) (where x and y both
are perfect squares of integers) on the parabola
y2 = px, p being a prime number is
(a)
one
(b)
zero
(c)
two
(d)
infinite
The equation of parabola whose focus is (0, 0) and
tangent at the vertex is x – y + 1 = 0 will be
(a)
x2 + y2 + 2xy – 4x + 4y – 4 = 0
(b)
x2 + y2 – 2xy – 4x + 4y – 4 = 0
(c)
x2 + y2 – 2xy – 4x – 4y + 4 = 0
(d)
none of these
The equation of the parabola with latus rectum
joining the points (3, 6) and (3, –2) will be
4a cosec  cot  (b)
4a tan  sec 
(a)
(y – 2)2 = 8(x – 1)
(c)
4a cos  cot 
4a sin  tan 
(b)
(y – 2)2 = 8(x – 5)
(c)
both (a) and (b)
(d)
none of these
The two parabolas y2 = 4x and x2 = 4y intersect at a
point P, whose abscissa is not zero, such that
(a)
they both touches each other at P
(b)
they cut at right angles at P
(c)
the tangents to each other at P make
complementary angles with the x-axis
(d)
none of the above
Einstein Classes,
13.
14.
none of these
2
(a)
(d)
4
The directrix of the parabola x = t 2 + 1 and
y = 2t + 1 will be
(a)
x=0
(b)
y=0
(c)
x=1
(d)
y=1
The area of the triangle formed by the lines joining
the vertex of the parabola x2 = 12y to the ends of its
latus rectum is
(a)
12 sq. unit
(b)
16 sq. unit
(c)
18 sq. unit
(d)
24 sq. unit
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 9
15.
16.
17.
18.
The point on the parabola y2 = 8x at which the
normal is parallel to the line x – 2y + 5 = 0
b
 
a
1/ 3

3
2
then the angle of
(b)
1

 ,  2
2


intersection of the parabola y2 = 4ax and x2 = 4by
at a point other then the origin is
(c)
1

 2,  
2

(d)
1

  2, 
2

(a)

(c)
=0
The straight line y = 2x +  does not meet the
parabola y2 = 2x if
23.
(a)

1
4
(b)
=5
(c)
 = –4
(d)
 = –1
The locus of the foot of perpendicular drawn to the
tangent of parabola y2 = 4ax from focus is
24.
(a)
x=0
(b)
y=0
(c)
y2 = 2a(x + a)
(d)
x2 + y2 (x + a) = 0
25.
The angle between the tangents drawn from the
origin to the parabola y2 = 4a(x – a) is
900
tan 1
(b)
1
2
(d)
300
450
26.
The angle between the tangents drawn at the end
points of the latus rectum of parabola y2 = 4ax is
(a)

3
(b)
2
3
(c)

4
(d)

2
The centre of the circle passing through the point
(0, 1) and touching the curve y = x2 at (2, 4) is
(a)
(c)
21.
1/ 3
 1 
  , 2
 2 
(c)
20.
a
If  
b
(a)
(a)
19.
22.
 16 27 
 , 
 5 10 
 16 53 
 , 
 5 10 
(b)
(d)
27.
28.
 16 5 
 , 
 7 10 
none of these
29.
If the segment intercepted by the parabola y2 = 4ax
with the line lx + my + n = 0 subtends a right angle
at the vertex then
(a)
4al + n = 0
(b)
nal + 4am + n = 0
(c)
4am + n = 0
(d)
al + n = 0
Einstein Classes,
30.

3
(b)


6
(d)


2
A line PQ meets the parabola y2 = 4ax in R such
that PQ is bisected at R. If the coordinates of P are
(x1, y1). Then the locus of Q will be
(a)
ellipse
(b)
hypebrola
(c)
circle
(d)
parabola
The curve described parametrically by x = t2 + t + 1
and y = t2 – t + 1 represents
(a)
parabola
(b)
hyperbola
(c)
ellipse
(d)
circle
The parabola y2 = 8x and the circle x2 + y2 = 2 have
number of common tangents
(a)
0
(c)
2
(b)
1
(d)

2
The focal chord to y = 16x is tangent to
(x – 6)2 + y2 = 2 then the possible values of the slopes
of this chord are
(a)
0
(b)
±1
(c)

(d)
tan–1 (3/2)
The equation of the parabola with (–3, 0) as a focus
and x + 5 = 0 as a directrix will be
(a)
y2 = 4(x – 4)
(b)
y2 = 4(x + 4)
(c)
x2 = 4(y + 4)
(d)
x2 = 4(y – 4)
The line x – 1 = 0 is the directrix of the parabola
y2 – kx + 8 = 0, then one of the values of k is
(a)
k=8
(b)
k = –8
(c)
kR
(d)
no real value of k
If x + y = k is a normal to the parabola y2 = 12x then
k is
(a)
k=9
(b)
(c)
kR
(d)
k=1
no real k
2
The point on the parabola y = 8x at which the
normal is inclined at 60 0 to the x-axis has the
coordinates
(a)
(6, 43)
(b)
(6, –43)
(c)
(43, 6)
(d)
(43, –6)
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 10
31.
The tangents from (0, a) to the parabola
y2 + 4a2 = 4ax are inclined at
(a)
(c)
32.
33.
0
(b)

6
(d)

3

2
The angle subtended at the focus by the normal
chord at the point (, ),   0 on the parabola
y2 = 4ax is
(a)
0
(c)

3
34.
(b)

6
(d)

2
35.
The locus of the mid point of the line segment
joining the focus to a moving point on the parabola
y2 = 4ax is another parabola with directrix
(a)
x = –a
(b)
x
a
2
(c)
x=0
(d)
x
a
2
If P 1Q 1 and P 2Q 2 are two focal chords of the
parabola y2 = 4ax, then the chords P1P2 and Q1Q2
intersect on the
(a)
directrix
(b)
axis
(c)
tangent at the vertex
(d)
none of these
Axis of a parabola is y = x and vertex and focus are
at a distance 2 and 22 respectively from the
origin. Then equation of the parabola is
(a)
(x – y)2 = 8(x + y – 2)
(b)
(x + y)2 = 2(x + y – 2)
(c)
(x – y)2 = 4(x + y – 2)
(d)
(x + y)2 = 2(x – y + 2)
ANSWERS (SINGLE CORRECT CHOICE TYPE)
1.
c
9.
c
17.
a
25.
c
33.
a
2.
a
10.
b
18.
a
26.
b
34.
c
3.
c
11.
a
19.
d
27.
b
35.
a
4.
b
12.
c
20.
c
28.
b
5.
b
13.
a
21.
a
29.
a
6.
b
14.
c
22.
a
30.
b
7.
d
15.
b
23.
d
31.
d
8.
c
16.
b
24.
a
32.
d
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 11
EXCERCISE BASED ON NEW PATTERN
COMPREHENSION TYPE
6.
The focus of the parabola will be
Comprehension-1
1.
2.
3.
The focus of any parabola is (1, 1) and the equation
of tangent at the vertex of parabola is given by
x + y = 1 then
(a)
 25 100 
 ,

 43 129 
(b)
 25 100 
,


 43 129 
The equation of the parabola will be
(c)
 43 129 
 ,

 25 100 
(d)
 43 129 
,


 25 100 
(a)
x2 + y2 – 2xy – 4x – 4y + 4 = 0
(b)
x2 + y2 + 2xy – 4x – 4y + 4 = 0
(c)
x2 + y2 + 2xy – 4x – 4y – 4 = 0
(d)
x2 + y2 – 2xy – 4x – 4y – 4 = 0
7.
The equation of the directrix of the parbaola will
be
(a)
4x  3y 
53
0
4
(b)
4x  3y 
53
0
4
(c)
4x  3y 
53
0
4
(d)
4x  3y 
53
0
4
The length of the latus rectum of this parabola will
be
(a)
2
(b)
22
(c)
32
(d)
42
The area of the triangle formed by latus rectum
and the tangents at the vertices of latus rectum
drawn from the point of intersection of directrix
and axis of parabola will be
(a)
0
(b)
1
(c)
2
(d)
3
8.
The vertex of the parabola will be
Comprehension-2
4.
Any parabola whose axis is parallel to y-axis and
the parabola is passing through the points
(0, 4), (1, 9) and (–2, 6) then
(a)
 48 36 
 ,

 25 25 
(b)
 48 36 
, 

 25 25 
The length of latus rectum of this parabola will be
(c)
 48 36 
, 

 25 25 
(d)
 48 36 
 , 
 25 25 
(a)
(c)
5.
1
2
(b)
5
2
(d)
3
2
7
2
Comprehension-4
Let the normals are drawn from any point (c, 0) to
the parabola y2 = x
9.
Angle between the tangents of parabola drawn from
11
the any point on the line y =
to the vertices of
4
focal chord is
10.
The maximum number of normals can be drawn
from the point (c, 0) to the parabola y2 = x is
(a)
2
(b)
3
(c)
4
(d)
5
The condition on c for which more than one normal can be drawn
(a)
0
(b)

3
(a)
c=0
(b)
c>
(c)

6
(d)

2
(c)
c<
1
2
(d)
All values of c
Comprehension-3
11.
The equation of the some conics are given
The value of c for which two normals are perpendicular to each other
7x2 + 2xy + 7y2 + 10x – 10y + 7 = 0
7x2 – 2y2 + 12xy – 2x + 9y – 22 = 0
1
2
(a)
c
1
4
(b)
c
5
4
(c)
c
7
4
(d)
c
3
4
9x2 – 24xy + 16y2 – 20x – 15y – 60 = 0
In the given conics one conic represents the
equation of parabola then
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 12
Comprehension-5
(D)
Any point which divides a chord of slope 2 of the
parabola y2 = 4ax internally in the ratio 1 : 2 then
12.
13.
The locus of the point which divides a chord of
slope 2 of the parabola y2 = 4x internally in the ratio
1 : 2.
(a)
ellipse
(b)
hyperbola
(c)
parabola
(d)
none
(c)
14.
 a 8a 
 , 
3 9 
(b)
  a 8a 
, 

 3 9 
(d)
 a  8a 
 ,

3 9 
  {7/4}
In column A the equation of parabola and its point
of contact with normal is given. In Column B the
equation of normals are given. Match the column
A with suitable equations of normals in column B.
Column - A
3 9 
 , 
 a 8a 
(s)
Matching-3
The focus of this conic will be
(a)
The given line neither
intersect nor touch the
parabola if possible
value of  is
(A)
2
Parabola y = 4ax
Column - B
(p)
and point of contact
Normal is
y = mx – 2a –
(am , –2am)
(B)
Parabola y2 = –4ax
(q)
point of contact
(a)
(c)
(b)
  2a  8a 
,


9 
 9
(d)
 2a  8a 
 ,

9 
 9
Normal is
(C)
(D)
Parabola x2 = 4ay
(r)
Parabola x2 = –4ay
(s)
 2a  a 
point of contact  2 , 2 
m m 
Match with the suitable parameters of the given
parabola 4x2 – 4x – 2y + 3 = 0.
Normal is
y = mx – 2am – am3
MULTIPLE CORRECT CHOICE TYPE
Column - B
1.
Consider a circle with its centre lying on the focus
of the parabola y2 = 2px such that it touches the
directrix of the parabola. Then a point of
intersection of the circle and the parabola is
Latus rectum
(p)
y=1
(B)
Directrix
(q)
y
9
8
(C)
Axis
(r)
y
7
8
(a)
P 
 ,P
2 
(b)
P

 , P
2


(D)
Tangent at vertex
(s)
x
1
2
(c)
 P 
 ,P
 2 
(d)
 P

  , P
 2

Matching-2
The straight line is y = 2x +  and the parabola is
given y2 = 2x. Match column A with suitable values
of column B.
Column - A
Column - B
The given line intersect
the parabola if possible
value of  is
(p)
(B)
The given line does not
intersect the parabola if
possible value of  is
(q)
  {–1/4}
(C)
The given line touches
the parabola if
possible value of  is
(r)
  {3/4}
Einstein Classes,
m2
Normal is
(A)
(A)
a
2a a 
point of contact  
, 2  y = mx + 2am + am3
 m m 
Matching-1
Column - A
y = mx + 2a +
(–am2, 2am)
 2a 8a 
 , 
 9 9 
MATRIX-MATCH TYPE
m2
2
The vertex of the conic is
  2a 8a 
, 

9 
 9
a
2.
  {1/4}
3.
If P is a point which moves in x-y plane such that
the point P is nearer to the centre of a square than
any of the sides. The four vertices of the square are
(±a, ±a). The region in which P will move is bounded
by parts of parabola of which one has the
equation.
(a)
y2 = a2 + 2ax
(b)
x2 = a2 + 2ay
(c)
y2 + 2ax = a2
(d)
x2 + 2ay = a2
The figure shows the parabola y = ax2 + bx + c then
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 13
(a)
(B)
If a chord of parabola y = 4x touches the parabola
y2 = 8x, then
Statement-1 is True, Statement-2 is True;
Statement-2 is NOT a correct
explanation for Statement-1
(C)
Statement-1 is True, Statement-2 is False
(a)
The tangent at the extremities of the
chord meet on the parabola y2 = x
(D)
Statement-1 is False, Statement-2 is True
(b)
The normals at the extremities of the
chord meet on the curve y2 = 2(x – 2a)3
(c)
The tangent at the extremities of the
chord meet on the parabola is y = 2x2
(c)
4.
(b)
b<0
(d)
c>0
2
b – 4ac > 0
2
(d)
5.
a<0
1.
STATEMENT-1 : The circle through P, Q, R
passes through the vertex of the parabola.
The normals at the extremities of the
chord meet on the curve 8y2 = (x – 2a)3
STATEMENT-2 : The centre of the above circle
Normals at the points P, Q, R on the parabola
y2 = 4ax meet in the point (h, k), then
(a)
 h  2a k 
is 
, .
4
 2
the centroid of the triangle PQR is
2.
 2(h  2a) 
,0

3


(b)
the centroid of the triangle PQR is
STATEMENT-1 : The
perpendicular to each other.
3.
the orthocentre of the triangle PQR is
k

 h  6a, 
2

4.
Each question contains STATEMENT-1 (Assertion)
and STATEMENT-2 (Reason). Each question has
4 choices (A), (B), (C) and (D) out of which ONLY
ONE is correct.
(A)
Statement-1 is True, Statement-2 is True;
Statement-2 is a correct explanation
for Statement-1
are
STATEMENT-1 : Two parabola y 2 = 4ax and
y2 = 4c(x – b) cannot have a common normal, other
than the axis.
STATEMENT-2 :
Assertion-Reason Type
tangents
STATEMENT-2 : Mutually perpendicular
tangents to the parabola meet on the line
2y + 3 = 0
the orthocentre of the triangle PQR is
k

 h  6a,  
2

(d)
Two tangents to the parabola x2 = 6y meet at the
3

point   1,   .
2

 h  2a 
, 0

 3

(c)
P, Q, R form a set of three co-normal points on
the parabola y2 = 4ax. The point of concurance of
the normals at P, Q, R is (h, k).
b
2.
ac
STATEMENT-1 : A point can be found such that
the two tangents from it to be parabola y2 = 4ax
are normals to the parabola x2 = 4by.
STATEMENT-2 : a2 > 8b2
5.
On the parabola y2 = 4ax, three points E, F, G are
taken.
STATEMENT-1 : The tangents at E and G
intersect on the ordinate of F.
STATEMENT-2 : The ordinates of E, F and G are
in G.P.
(Answers) EXCERCISE BASED ON NEW PATTERN
COMPREHENSION TYPE
1.
a
2.
b
3.
c
4.
a
5.
7.
a
8.
c
9.
b
10.
b
11.
13.
a
14.
d
MATRIX-MATCH TYPE
1.
[A-q; B-r;C-s; D-p]
2.
[A-q; B-r, s; C-p; D-q, r, s]
MULTIPLE CORRECT CHOICE TYPE
1.
a, b
2.
a, b, c, d
3.
a, b, c, d
4.
ASSERTION-REASON TYPE
1.
A
2.
A
3.
A
4.
A
5.
Einstein Classes,
d
d
6.
12.
d
c
3.
[A-s; B-r; C-q; D-p]
a, b
5.
a, c
A
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 14
INITIAL STEP EXERCISE
(SUBJECTIVE)
1.
2.
Find the equation of the common tangents of
(a)
the parabolas y2 = 4ax & x2 = 4by
(b)
the circle x2 + y2 = 4ax & y2 = 4ax
PN is an ordinate of the parabola; a straight line is
drawn parallel to the axis to bisect NP and meets
the curve in Q; prove that NQ meets the tangent at
the vertex in a point T such that AT 
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
15.
16.
2
NP .
3
If perpendiculars be drawn on any tangent to a
parabola from two fixed points on the axis, which
are equidistant from the focus prove that the
difference of their squares is constant.
Prove that the distance between a tangent to the
parabola and the parallel normal is a cosec sec2,
where  is the angle that either makes with the axis.
(parabola : y2 = 4ax).
From an external point P tangents are drawn to
the parabola ; find the equation to the locus of P
when these tangents make angle 1 and 2 with the
axis such that
(a)
tan1 + tan2 = b
(b)
tan1 tan2 = c
(c)
cot1 + cot2 = d
(d)
1 + 2 = const. = 2a
(e)
tan21 + tan22 = 
(f)
cos1 cos2 = µ
Find the locus of the middle points of chords of the
parabola which
(a)
pass through the focus.
(b)
pass through the fixed point (h, k)
(c)
are normal to the curve.
(d)
subtend a constant angle ‘’ at the
vertex.
Show that the locus of the point that divides a chord
of slope 2 of the parabola y2 = 4ax internally in the
ratio 1 : 2 is a parabola.
Perpendicular chords OP and OQ are drawn
through the vertex O of the parabola y2 = 4ax. Prove
that for all positions of P, PQ intersects the axis at a
fixed point. Find also the locus of the middle points
of PQ.
Prove that the line lx + my + n = 0 touches the
parabola y2 = 4ax if ln = am2.
Prove that the chord of the parabola y2 = 4ax, whose
equation is y – x2 + 4a2 = 0 is a normal to the
curve & that its length is 63 a.
Prove that the normal chord at the point whose
abscissa is equal to its ordinate subtends a right
angle at the focus.
Prove that for all values of , the line y = (x – 11)
cos  – cos 3 is a normal to the parabola y2 = 16x.
Tangents to the parabola y2 = 8x make an angle of
450 with the straight line y = 3x + 5. Find the
equation of tangents.
Einstein Classes,
14.
17.
Find the vertex, directrix & focus of the parabola
(x + y – 1)2 = 8 (2x + y – 3).
Find the length of the normals drawn from the point
on the axis of the parabola y2 = 8ax whose distance
from the focus is 8a.
Consider a circle with centre lying at the focus of
the parabola y 2 = 2px such that it touches the
directrix of the parabola. Find the points of
intersection of the circle and the parabola.
A parabola is drawn to pass through A and B, the
ends of a diameter of a given circle of radius a, and
have as directrix a tangent to a concentric circle of
radius b; the axes being AB and a perpendicular
diameter, prove that the locus of the focus of the
x2
y2
 1.
parabola is 2  2
b
b  a2
18.
19.
20.
21.
If a2 > 8b2, prove that a point can be found such
that the tangents from it to the parabola
y2 = 4ax are normals to the parabola x2 = 4by.
Find the locus of
(a)
the point of intersection of tangents to the
parabola y2 = 4ax drawn at the end
points of focal chords.
(b)
the locus of the middle points of chords
of the parabola which subtend a right
angle at the vertex, and prove that these
chords have their mid-points on the
parabola.
Find the locus of a point which is such that (a) two
of the normals drawn from it to the parabola are at
right angles. (b) the three normals through it cut
the axis in points whose distances from the vertex
are in arithmetical progression.
Tangents are drawn at any point (x1, y1) on the
parabola y2 = 4ax. Now tangents are drawn from
any point on this tangent to the circle x2 + y2 = a2
such that all the chords of contact pass through a
x
y 
fixed point (x2, y2). Prove that 4 1  1   0 .
 x2 y 2 
22.
23.
24.
If from the vertex of a parabola a pair of chords be
drawn at right angles to one another and with these
chords as adjacent sides a rectangle be made, prove
that the locus of the point diagonally opposite to
vertex of the rectangle is the parabola
y2 = 4a(x – 8a)
Find the equation of a circle which touches the
y-axis and the parabolas y2= x and 2y = 2x2 – 5x + 1
only at the point where they touch each other and
at no other point.
Prove that the locus of the mid points of all the
tangents drawn from points on the directrix to the
parabola y2 = 4ax is y2(2x + a) = a(3x + a)2.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 15
25.
Show that the locus of the points of intersection of
tangents to y2 = 4ax, which intercept a constant
length d on the directrix is (y2 – 4ax) (x + a)2 = d2x2.
26.
Show that the locus of a point such that two of the
three normals drawn from it to the parabola
y2 = 4ax coincide is 27ay2 = 4(x – 2a)3.
FINAL STEP EXERCISE
(SUBJECTIVE)
1.
2.
3.
Prove that the locus of the middle point of the
portion of a normal intercept between the curve &
the axis is another parabola whose vertex is the focus and whose latus rectum is one quarter of that
of the original parabola.
Prove that the area of the triangle formed by three
points on a parabola is twice the area of the
triangle formed by the tangents at these points.
For a parabola y2 = 4ax if tangents at the points
(x’, y’) & (x’’, y’’) meet at the point (x1, y1) and
normals at the same points in (x2, y2). Prove that
5.
6.
7.
x1 
(ii)
x 2  2a 
y 2  y y   y  2
4a
14.
If a chord which is normal to the parabola at one
end subtends a right angle at the vertex, prove that
it is inclined at an angle tan –12 to the axis.
(parabola : y2 = 4ax)
From the point where any normal to the parabola
y 2 = 4ax meets the axis is drawn a line
perpendicular to this normal. Prove that this line
always touches an equal parabola.
Tangents are drawn to the parabola y2 = 4ax at
points whose abscissae are in the ratio µ : 1 ; prove
that they intersect on the curve y2 = (µ1/4 + µ–1/4)2 ax.
Prove that the parabolas y2 = 4ax & x2 = 4ay cut
 3(ab )1 / 3
one another at an angle tan  2 / 3
2/ 3
 2a  b
9.
12.
13.
( y   y  )
8a 2
1 
8.
11.
y y 
y   y 
and y1 
4a
2
(i)
y 2   y y 
4.
10.
15.
16.
17.
18.

.


a2
x y 
and the parabola y2 = 4ax. Let these
2
2
If a normal to a parabola makes an angle  with
the axis, show that it will cut the curve again at an
1

angle tan 1  tan   .
2


Prove that the two parabolas y2 = 4ax and y2 = 4c
(x – b) cannot have a common normal other than
the axis, unless
b
 2.
ac
19.
20.
21.
Einstein Classes,
PNP’ is a double ordinate of the parabola ; prove
that the locus of the point of intersection of the
normal at P and the straight line through P’
parallel to the axis is the equal parabola y2 = 4a
(x – 4a) .
The normal at any point P meets the axis in G and
the tangent at the vertex in G’. If A be the vertex
and the rectangle AGQG’ be completed, prove that
the equation to the locus of Q is x3 = 2ax2 + ay2.
Two equal parabolas with axes in opposite
directions touch at a point O. From a point P on
one of them are drawn tangents PQ and PQ’ to the
other. Prove that QQ’ will touch the first parabola
in P’ where PP’ is parallel to the common tangent
at O.
Two tangents to a parabola meet at an angle of 450.
Prove that the locus of their point of intersection is
the curve y2 – 4ax = (x + a)2.
Show that the locus of the point of intersection of
two tangents which with the tangent at the vertex
form a triangle of const. area C 2 is
x2(y2 – 4ax) = 4C4.
Through each point of the straight line x = my + h
is drawn the chord of the parabola y2 = 4ax which
is bisected at the point. Prove that it always touches
the parabola (y + 2am)2 = 8a(x – h).
Find the locus of a point such that slopes m1m2m3
of the normals from it to the parabola y2 = 4ax are
related by tan–1m12 + tan–1m22 + tan–1m32 = .
If the three normals drawn at three different points
on the parabola y2 = 4ax pass through the point
(h, k), then prove that h > 2.
Common tangents are drawn to the circle
2
common tangents intersect at the point A. Find the
area of the quadrilateral formed by the common
tangents and the chords of contact of the point A
w.r.t. the circle & the parabola.
If a circle passes through the feet of the normals
drawn from a point, prove that the circle also passes
through the origin.
If P, Q, R be the foot of the normals to the parabola
y2 = 4ax that pass through the point T, then prove
that SP . SQ . SR = a ST2.
Find the shortest distance between y2 = 4ax and the
circle x2 + y2 – 24y + 128 = 0.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MP – 16
ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE)
1.
5.
6.
8.
13.
15.
20.
(a)
b1/3y + a1/3x + a2/3 b2/3 = 0
(b)
x=0
(a)
y = ad
(b)
cx = a
(c)
y=d
(d)
y = (x – a) + tan 2a
(e)
y2 – x2 = 2ax
(f)
x2 = µ2 [(x – a)2 + y2]
(a)
y2 = 2a(x – a)
(b)
y2 – ky = 2a(x – h)
(c)
y2(y2 – 2axd + 4a2x) + 8a4 = 0
(d)
8a2 + y2 – 4x2 + tan2a = 16a2 (4ax – y2)
2ax = y2 + 8a2
y + 2x + 1 = 0; 2y = x + 8
9; 10a
(a)
y2 = a(x – 3a)
(b)
27ay2 = 2(x – 2a)3
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111