COORDINATE GEOMETRY
NOTES
(x 2  x1 )2  (y 2  y1 )2
1.
If A(x1, y1) and B(x2, y2) are two points then AB =
2.
 mx 2  nx1 my 2  ny1 
If P divides AB in the ratio m : n then P = 
,
.
mn 
 mn
3.
 x  x 2 y1  y 2
Mix point of AB is  1
,
2
 2

.

4.
A(x1, y1), B(x2, y2), C(x3, y3) are the vertices of a triangle ABC then the centroid G =
 x1  x 2  x 3 y1  y 2  y 3 
,

.
3
3


5.
The line joining a vertex of a triangle to the mid point of the opposite side is called a median.
The medians of a triangle are meeting at a point, called the centroid G of the triangle.
y  y1
The slope of the line joining the points (x1, y1) and (x2, y2) is m = 2
.
x 2  x1
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
If a line makes an angle  with the positive X-axis in the positive direction, then its slope
m = tan  .
a
The slope of the line ax + by + c = 0 is m =
(if b  0)
b
Let m1 and m2 be the slopes of two nonvertical lines L1 and L2. If L1 || L2 then m1 = m2 and if L1
 L2 then m1m2 = -1.
The equation the line with slope m and intercept on Y – axis as c is y = mx + c.
The equation of the line passing through (x1, y1) and having slope m is y – y1 = m(x – x1).
The equation of the line passing through two points (x1, y1) and (x2, y2) is
(x – x1)(y2 – y1) = (y – y1)(x2 – x1)
x y
If a line makes intercepts a and b on the coordinate axes, then its equation is  = 1.
a b
x y
1
The area of the triangle formed by the line  = 1 with the coordinate axes is   |ab|.
a b
2
The general form of the equation of a line is ax + by + c = 0 (a2 + b2  0).
Slope of X - axis is 0 ; Slope of a horizontal line is 0
Slope of Y – axis is not defined
Slope of a vertical line is not defined
Equation of X – axis is y = 0.
Equation of the horizontal line passing through (h, k) is y = k.
Equation o f Y – axis is x = 0.
Equation of the vertical line passing through (h, k) is x = h.
To find the point of intersection of two lines, solve the two equations for x and y.
If three lines are meeting at one point then the lines are concurrent and the point is called the
point of concurrence of the three lines.
The equation of a line parallel to ax + by + c = 0 is in the form ax + by + K = 0.
The equation of a line perpendicular to ax + by + c = 0 is in the form bx – ay + K = 0.
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Coordinate Geometry
WORKED EXAMPLES
1.
If A(2, 1), B(-1, -2), C(0.4), P(x, y) and PA2 + PB2 = 2 PC2 then find the value of x + 9y
2
2
2
2
2
Sol: (x – 2) + (y – 1) + (x + 1) + (y + 2) = 2[(x – 0) + (y – 4) ]
2
2
2
2
2
2
 x – 4x + 4 + y – 2y + 1 + x + 2x + 1 + y + 4y + 4 = 2x + 2y + 32 – 16y
 -2x – 18y – 22 = 0 x + 9y = -11
2.
If the point of intersection of the lines x + 3y – 1 = 0 and x – 2y + 7 = 0 lies on the line
x + Ky – 16 = 0, then find K.
Sol:
x + 3y
x – 2y
– +
5y
- 1=0
+7=0
–
- 8=0
8
5
24
19
x +
-1 = 0  x =
5
5
 19 8 
 point of intersection is 
, 
 5 5
y=
19 8K
99

 16 = 0  -19 + 8K – 80 = 0  K =
5
5
8
3.
Find the point on the line x + y + 7 = 0 equidistant from (0, 1), (2, 1)
Sol: Let the point be P(x, y)
Then x + y + 7 = 0 ….. (1)
(x – 0)2 + (y – 1)2 = (x – 2)2 + (y – 1)2
2
2
2
2
 x + y – 2y + 1 = x – 4x + 4 + y – 2y + 1  4x = 4  x = 1
from (1), 1 + y + 7 = 0 and so, y = -8
 P = (1, -8).
4.
Find the equation of the line passing through the point of intersection of the lines
2x – y = 0 and x + 2y – 5 = 0 and perpendicular to the line 3x – 2y + 4 = 0
Sol: 2x – y = 0
….. (1)
x + 2y – 5 = 0
……(2)
3x – 2y + 4 = 0 ……(3)
(1) x 2 gives
4x – 2y = 0
x + 2y – 5 = 0 (From (2))
5x - 5 = 0
x=1
y=2
 P = (1, 2)
A line perpendicular to (3) is 2x + 3y + K = 0
2 + 6 + K = 0  K = -8
Hence, the required line is 2x + 3y – 8 = 0
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5.
Find the distance of the point (2, 1) from the line 3x + 2y – 18 = 0 measured along the line
x – y – 1 = 0.
Sol: By solving the two equations,
We get B = (4, 3)
Let A = (2, 1)
 AB =
2
3x
2
(4  2)  (3  1)  8
=
18
y2
+
B
(4,3)
0
X-y-1=0
X
A (2,1)
6.
D (2, 1), E (-1, -2) and F(3, 3) are the midpoints of the sides BC, CA, AB of a triangle
ABC. Then find the vertices of triangle ABC.
Sol: Let A = (h, k).
A
Then AFDE is a parallelogram and so,
E (-1,-2)
(3,3) F
mid point of AD = mid point of EF.
 h  2 k  1  3  1 3  2 

,
,


2   2
2 
 2
B
h = 1, k = -3 and A = (1, -3)
Similarly, B = (6, 6) and C = (-2, -4)
D
(2,1)
C
7.
Find the equation of the line which passes through (3, 4) and is such that the portion of it
intercepted by the coordinate axes is divided by the point in the ratio 2 : 3
Sol: From the figure
Y
 3a  0 2b  0 
P= 
,
  (3,4)
B
3
(0,b)
 23 23 
P (3,4)
2
 a = 5, b = 10
 Equation of the line is
x y

 1 i.e., 2x + y = 10
5 10
8.
O
A
(a,0)
X
ABCD is a square. The coordinates of opposite vertices A and C are (1, 3), (3, 7). Then
find the equation of BD.
D
Sol: P = mid point of AC
 1 3 3  7 
= 
,
 = (2, 5)
2 
 2
73 4
Slope of AC =
 2
3 1 2
1
 Slope of BD is m =
2
C (3,7)
900
P
A
(1,3)
B
1
(x – 2)
2
i.e., 2y – 10 = -x + 2 or x + 2y – 12 = 0.
 Equation of BD is y – 5 =
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LEVEL – I
1.
Show that the points (e,  ), (2e, 3  ), (3e, 5  ) are collinear.
2.
The points (3, 2), (-3, 2), (0, h) are the vertices of an equilateral triangle. If h < 2, find h.
3.
A (-4, 3), B (6, -1) and C (2, -5) are the vertices of a triangle. Find the length of the median
through A.
Find the condition that the points (7, 5), (3, 6), (x, y) to be collinear.
A triangle with vertices (4, 0), (-1, -1), (3, 5) is …………….
(AIEEE 2002).
1
The point P, if Q = (-2, 4) is the point on the line segment OP such that OQ = OP is ……….,
3
O being the origin.
4.
5.
6.
7.
8.
9.
10.
11.
12.
If the point P(x, y) be equidistant from the points A(a + b, a – b) and B(a – b, a + b) then show
that x=y.
If A(1, 2), B(2, -3), C(-2, 3), P(x, y) and PA2 + PB2 = 2PC2, then find the value of 7x – 7y.
A square has two opposite vertices at (2, 3) and (4, 1) then find the length of the side.
If (1, 1), (3, 4) are two adjacent vertices of a parallelogram with center (5, 4) then find the
remaining vertices.
3 5
If the mid point of join of (x, y + 1) and (x + 1, y + 2) is  ,  , then find the mid point of join of
 2 2
(x – 1, y + 1) and (x + 1, y – 1).
One vertex of a rectangle of sides 3 and 4 lies at the origin. The coordinates of the opposite
vertex may be (3, 4). (True/False).
13.
Show that the points (1, -1), ( 3 , 3 ), (0,
triangle.
14.
The medians AD and BE of  ABC with vertices A(o, b), B(0,0), C(a,0) are mutually
perpendicular if a=
Find the value of K such that the line 3x + 4y + 5 – K (x + 2y + 3) = 0 is parallel to Y – axis.
If (a, b) is the middle point of the intercept of a line between the coordinate axes, then find the
equation of the line.
Find the angle subtended by the line joining the points (5, 2) and (6, -15) at (0, 0).
If the area of the triangle formed by the lines x = 0, y = 0, 3x + 4y = a (a > 0) is 1 then find a.
A(-1, 1), B(5, 3) are opposite vertices of a square. Find the equation of the other diagonal (not
passing through A, B).
If the point of intersection of the lines x – 3y + 5 = 0 and Kx + 4y + 2 = 0 lies on the line
2x + 7y – 3 = 0, then find K.
15.
16.
17.
18.
19.
20.
3 -1) are the vertices of a right angled isosceles
LEVEL – II
1.
2.
3.
4.
Prove that the points (-3, 1), (-2, -3), (2, -2), (1, 2) form a square.
Prove that the points (-a, -b), (0, 0), (a, b), (a2, ab) are collinear.
One end of a line segment of length 10 is (2, -3). If the other end has the abscissa 10 then find
its ordinate.
Find the equation of the straight line which bisects the intercepts made by the lines 2x +3y = 6,
3x +2y = 6 on the axes
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6.
1
1
 1 2 
If S = (t2, 2t) and S1 =  2 ,  and A = (1, 0), then find the value of
+
.
AS AS1
t t 
Show that the points (,),( , ),(, ),( , ) where , , ,  are different real numbers are
7.
vertices of a square.
If two vertices of an equilateral triangle are (0, 0), (3,
5.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
3 ), find the third vertex.
Find the intercept on Y – axis of the line joining the points (2, 3) and (-4, -2).
2
2
The line joining the points (p + 1, 1) and (2p+1, 3) passes through the point (2p , -2p ) for a
value of p = …………
Find the point on the line 2x – 3y = 5 equidistant form (1, 2), (3, 4).
The triangle formed by the lines x + y = 0, 3x + y – 4 = 0, x + 3y – 4 = 0 is ……………
(-4, 5) is a vertex of a square and one of these diagonals is 7x – y + 8 = 0. Find the equation of
the other diagonal.
A line passing through A(1, -2) has slope 1. Find the points on the line at a distance of 4 2
units form A.
Find the equation of the line joining the origin to the point of intersection of the lines
x y
x y
  1 and   1 .
3 4
4 3
If the lines y = 4 – 3x, ay = x + 10, 2y + bx + 9 = 0 represent three consecutive sides of a
rectangle, then find the value of ab.
Find the equation of the line passing through the point of intersection of the lines x – 2y + 5 = 0
and 3x + 2y + 7 = 0 and perpendicular to the line x – y = 0.
If a straight line perpendicular to the line 2x – 3y + 7 = 0 forms a triangle with the coordinate
axes whose area is 3 sq. units, then find the equation of the line.
(EAMCET 2002).
Find the distance of the point (2, 3) from the line 2x – 3y + 9 = 0 measured along the line
x – y + 1 = 0.
If 2x + 3y + 4 = 0 is the perpendicular bisector of the segment joining the points A(1, 2) and
B (, ) the find the value of    .
The points (1, 3) and (5, 1) are two opposite vertices of a rectangle. The other two vertices lie on
the line y = 2x + c. Then find the remaining vertices.
LEVEL – III
1.
2.
3.
4.
5.
6.
A line AB is divided internally at P(2, -4) in the ratio 2 : 3 and at Q (0, 5) in the ratio 3 : 2. Then
find A.
Find the point on the line 3x + y + 4 = 0 which is equidistant from the points (-5, 6) and (3, 2).
A line meets the X-axis at A and Y – axis at B such that centroid of triangle OAB is (1, 2). Then
find the equation of AB.
1
1
1
 a 2a 
Given A = (at2, 2at), B =  2 ,

 .
 and S = (a, 0); prove that
t 
SA SB a
t
D(2, 1), E(-1, -2) and F(3, 3) are the midpoints of the sides BC, CA, AB of triangle ABC. Find the
equation of the sides of triangle ABC.
ABCD is a parallogram. Equations of AB and AD are 4x + 5y = 0, 7x + 2y = 0 and the equation
of the diagonal BD is 11x + 7y – 9 = 0. Find the equation of AC.
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7.
8.
9.
10.
Coordinate Geometry
The equations of the lines cutting off intercepts a, b on the coordinate axes such that a + b = 3,
ab = -4 are ………..
Find the equation of the line which passes through (6, -10) and is such that the portion of it
intercepted by the coordinate axes is divided by the point in the ratio 5 : 2.
Find the equation of the line which passes through the point (3, 4) and the sum of its intercepts
on the axes is 14.
The equations of perpendicular bisectors of the sides AB and AC of a triangle ABC are
x – y + 5 = 0 and x + 2y = 0. If A = (1, -2) then find the equation of the side BC.
KEY TO LEVEL – I
2. 2 –
27
8. –4
14. 
2b
3. 10
4. x + 4y = 27
5. isosceles and right angled
9. 2
10. (7, 4), (9, 7)
15. 2
16.
19. 3x + y – 8 = 0
11. (1, 1)
x y
 2
a b
17.

2
6. (-6, 12)
12. True
18. 2 6
20. 3
KEY TO LEVEL – II
3. 3 or -9
8.
4
3
4. 2x + 2y –5 = 0
5. 1
7. (0, 2 3 )or (3, - 3 )
9. –2
10. (4, 1)
11. isosceles
15. 18
16. x + y + 2 = 0
13. (-3, -6), (5, 2) 14. x + y = 0
17. 3x + 2y +=  6
18. 4
2
19.
81
13
12. x + 7y – 31 = 0
20. (2, 0) and (4, 4)
KEY TO LEVEL – III
1. (6, -2) 2. (-2, 2)
6. x – y = 0
3. 2x + y – 6 = 0
5. 5x – 4y – 6 = 0, 2x – y = 0, x – y = 0
7. x – 4y = 4, 4x – y + 4 = 0
9. x + y = 7 or 4x + 3y = 24
8. 2x – 3y – 42 = 0
10. 14x + 23y – 40 = 0
WORK SHEET
1. Write the equations of straight lines (i) parallel to X –axis and at a distance of 4 and 5 units above
and below to X-axis respectively. (ii) Parallel to Y-axis and at a distance of 2 units, from Y-axis to
the right of it. (iii) at a distance of 6 units from Y-axis to the left of it.
2. The slope of the line joining points A  ab12 ,2ab1  ,B  ab22 ,2ab2  is……
3. The slope of the line perpendicular to the line joining the points A(a, 0), B(0, b) is ……..
b
4. Find the equation of a line passing through (a, b) and having slope .
a
5. If (2, 1), (–5, 7), (–5, –5) are the midpoints of the sides of a triangle. Find the equations of the
sides of the triangle.
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1
and whose (i) y intercept is 2 (ii) x – intercept is – 2.
2
7. Find the equation of a line passing through (5, 4) and parallel to 2x + 3y + 7 = 0
x y
8. Find the equation of a line passing through (b, a) and perpendicular to + = 1 .
a b
6. Find the equation of lines having slope
9.
10.
11.
12.
13.
14.
15.
A line intersects the coordinate axis such that the area of the triangle formed is 6 sq.units. Also
the sum of the intercepts made by the line is 5, then find the equations of line.
Find the equation of the line passing through (1, 1) and making equal intercepts on the
coordinate axes.
Find the area of the square bounded by the lines x + y =  2, x – y =  2.
Find the equation of the line passing through the intersection of the lines 2x – 5y + 1 = 0, 3x + 2y
= 8 and (i) making equal intercepts on the axes (ii) parallel to x + y – 2 = 0 (iii) perpendicular to
4x + 5y + 2 = 0.

Find the length of the altitude from A to BC of ABC when A = ( 5, –3), B = (–3, –2),
C = (9, 12).
If the area of the triangle formed by the lines x =0, y = 0 and 3x + 4y = a is 6 sq.units, then find
the value of a.
If A
(10,
4),
B(–4, 9), C(–2, –1) are the vertices of a triangle, then find the equation so
 
(i) AC, BC, CA
(ii) medians through A, B, C
(iii) The altitudes through A, B, C
(iv) Perpendicular bisectors of the sides AB, BC, CA.
KEY TO WORK SHEET
1) (i) y = 4, y = - 5 (ii) x = 2, (iii) x= – 6
4) bx – ay = 0
2
b 2 + b1
5) 6x + 7y + 65 = 0, x + 5 = 0, 6x – 7y + 79 = 0
7) 2x + 3y – 22 = 0
10) x + y – 2 = 0
2)
8)
x y
- =0
b a
3)
a
b
6) x+2y– 4=0, x–2y–2 =0
9) 3x + 2y – 6 = 0, 2x + 3y–6= 0
11) 8 sq.units
12) i) x + y – 3 = 0, ii) x + y – 3 = 0 iii) 5x – 4y – 6 =0
13)
62
85
units
14) a = 12
15)
(i) 5x + 14y – 106 = 0, 5x + y + 11 = 0, 5x – 12y – 2 = 0
(ii) y – 4 = 0, 15x + 16y – 84 = 0, 3x – 2y + 4 = 0
(iii) x – 5y + 10 = 0, 12x + 5y + 3 = 0, 14x – 5y + 23 = 0
(iv) 28x – 10y – 19 = 0, x – 5y + 21 = 0, 24x +10y – 111 = 0

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