Quadratic Equations and Conics
A quadratic equation in two variables is an equation that’s equivalent to
an equation of the form
p(x, y) = 0
where p(x, y) is a quadratic polynomial.
Examples.
• 4x2 3xy 2y 2 + x y + 6 = 0 is a quadratic equation, as are
x2 y 2 = 0 and x2 + y 2 = 0 and x2 1 = 0.
• y = x2 is a quadratic equation. It’s equivalent to y
and y x2 is a quadratic polynomial.
x2 = 0,
• xy = 1 is a quadratic equation. It’s equivalent to the quadratic
equation xy 1 = 0.
• x2 +y 2 =
1 is a quadratic equation. It’s equivalent to x2 +y 2 +1 = 0.
• x2 + y = x2 + 2 is a not a quadratic equation. It’s a linear equation.
It’s equivalent to y 2 = 0, and y 2 is a linear polynomial.
A conic is a set of solutions of a quadratic equation in two variables. In
contrast to lines–solutions of linear equations in two variables–it takes a fair
amount of work to list all of the possible geometric shapes that can possibly
arise as conics. In what remains of this chapter, we’ll take a tour of some
conics that we already know. After we cover trigonometry in this course,
we’ll return to conics and explain all of the possible shapes of conics.
154
J
No points
We saw in the chapter Polynomial Equations that the quadratic equation
x2 + y 2 =
1
has no solution. A sum of squares can never be negative. Thus, ; is an
example of conic.
x2 = 1 has no solution too!
One
ts iii the graph
of point
f(x) are points of the form (x, f(x)). That is,
in the graph have
theinform
(x, x
). Polynomial
2
These are the
points in
thethe quadratic equation
We saw
the chapter
Equations
that
e second coordinates are the square of theirxfirst
or
2
2
+ ycoordinates,
=0
ords, points of the form (x, y) where
has exactly one solution, the point (0, 0). The only way to take two numbers,
=
square them,y take
the sum of those squares, and get 0, is if the two numbers
the graph you
of f(x)
2 inare
x
the0.plane
is the
set as thethe
set single point (0, 0) is a
start= with
Thus,
the same
set containing
of the quadratic
conic. equation in two variables given by the equation
0
I
150
155
Parallel lines
lines
Parallel
An equation
equation for
for the
the vertical
vertical line
line that
that crosses
crosses the
the x-axis
x-axis atat the
the number
number 11
An
or equivalently,
equivalently, xx 11 = 0.0. Similarly,
Similarly, xx++11 ==00 isis an
anequation
equationfor
for
isis xx == 1,1, or
the vertical
vertical line
line that
that crosses
crosses the
the x-axis
x-axis atatthe
thenumber
number —1.1. As
Aswe
wesaw
sawininthe
the
the
previous chapter,
chapter, the
the union
union of
of these
these two
two vertical
vertical lines
lines isisthe
theset
set ofofsolutions
solutions
previous
of the
the equation
equation
of
(x 1)(x
1)(x++1)1)==00
(x
an equation
equation that
that can
can be
be written
written more
more simply
simply asas
an
—
—
x2—O
1=0
x
1
2
Thus, two
two parallel
parallel lines
lines cami
can appear
appear as
as aa conic.
conic.
Thus,
—1
Parallel lines
Intersecting
lines
the vertical line that crosses t.he x-axis at the number 1
An equation for
Intersecting
lines
equivalently,
isIt’s
x =also
1, possible
or
equation
0. Similarly,
for
= 0 is In
anthe
+ aa1 conic.
It’s
also
possible
to have
havextwo
two1intersecting
intersecting
linesxas
as
conic.
the
previous
to
lines
In
previous
huethat
thatthe
crosses
the vertical
the x-axis
at the number —1. As we saw in the
chapter
we saw
saw
that
the
solutions
of
chapter
solutions
of
we
previous chapter, the union of these
two vertical lines is the set of solutions
x2 y 2 == 00
of the equation
are aa union
union of
of the
the two
two lines
lines that
that
have
slope
and —1,1, and
and that
that each
each pass
pass
(x have
1)(x +
1) =110and
are
slope
through
the point
point
(0,0).
0).
Recall that
that
an equation
equation for
for the
the first
first ofof
through
the
(0,
xx simply
yy == 00asisis an
that can
be Recall
written
an equation
more
these two
two lines,
lines, and
and that
that xx++yy == 00 isis an
an equation
equation for
for the
the second
second of
ofthese
thesetwo
two
these
—1an=equation
0
lines. Therefore,
Therefore, (x
(x y)
y)(x
y) == 00 isis
for the
the union
union of
of these
thesetwo
two
lines.
(x ++y)
an equation for
2
2
lines,
and
can be
be lines
simplified
as x asya ==
Thus,and
can appear
twoititparallel
conic.
lines,
simplified
as
can
0.0.
—
—
—
—
—
—
/N
156
149
Intersecting lines
It’s also possible to have two intersecting lines as a conic. In the previous
chapter we saw that the solutions of
2
Single line
x2 = 0 is a quadratic equation, so its set of solutions in the plane is a conic.
The only number whose square is 0 is 0 itself, so if x2 = 0, then x = 0.
Thus, the set of solutions of x2 = 0 are those points in the plane whose xcoordinates equal 0. This is a single vertical line, the y-axis. This single line
is an example of a conic.
__ ___
_____
arabola
The graph ofParabola
the function in one variable f(x) = x
2 is called a parabola.
The graph of the function in one variable f (x) = x2 is called a parabola.
J(x=x2
The points in the graph of f (x) are points of the form (x, f (x)). That is,
have the
form
(x, (x,
x2 ).f(x)).
TheseThat
are is,
the points in the
The points inthe
thepoints
graphinofthe
f(x)graph
are points
of the
form
plane
whose
coordinates
the are
square
theirinfirst
points in the
graph
havesecond
the form
(x, x
). are
2
These
the of
points
the coordinates, or
in other
words, points
of the
formof(x,
y) where
coordinates
ne whose second
are the
square
their
first coordinates, or
other words, points of the form (x, y) where y = x2
=
157
x2 in the plane is the same set as the set
To recap, the graph of f(x)
solutions of the quadratic equation in two variables given by the equation
=
=
_______
_______
_______
Parabola
The graph of the function in one variable f(x)
=
x2
is called a
To recap, the graph of f (x) = x2 in the plane is the same set as the set
arabola of solutions of the quadratic equation in two variables given by the equation
2
= is
x2 aisconic.
= xfunction
. That is,
below
The graph ofy the
in the
called a parabola.
one parabola
variable f(x)
The points in the graph of f(x) are points of the form (x, f(x
the points in the graph have the form (x, x
). These are the p
2
plane whose second coordinates are the square of their first coo
in other words, points of the form (x, y) Iiere
The points in the graph of f(x) are points of the form (x, f(x)). That is,=
e points in the graph have the form (x, x
). These are the points in the
2
Hyperbola
the graph
of f(x) = x
To recap,
2orin the plane is the same se
ane whose second coordinates are the square
first coordinates,
of their
1
1
The
points
in
the
graph
of
the
function
g(x)
=
are
of
the
(x,variables
solutions of the quadratic
given by
equation form
in two
x
x ).
other words, points of the form (x, y)ofIiere
=
.
2
y=1
To recap, the graph of f(x) = x
2 in the plane is the same set as the set
solutions of the quadratic equation in two variables given by the equation
.
2
=1
J
)(s)
2
A(
a
That is, they are points whose y-coordinates are the multiplicative
inverse of
150
1
their x-coordinates. In other words, the graph of g(x) = x is exactly the set
of points (x, y) in the plane where y = x1 .
158
150
J
The points in the graph of f(x) are points of the form (x, f(x)
the points in the graph have the form (x, x
). These are the po
2
1
We know from the domain
the function
g(x) = x that
6= 0.
Thus,
plane of
whose
second coordinates
are xthe
square
of we
their first coo
1
can multiply the equation
y = x words,
by x topoints
obtainofthe
in other
the equivalent
form (x, y)equation
where
=
xy = 1
so that we now have that To
therecap,
graph the
of x1 graph
is exactly
the =
setx2of in
solutions
of the
of f(x)
the plane
is the same se
equation xy = 1.
of solutions of the quadratic equation in two variables given by th
=
J
150
The equation xy = 1 is a quadratic equation, so its set of solutions is a
conic. It’s called a hyperbola.
159
__
_______
_______
Exercises
#1-5 involve writing equations for planar transformations of the parabola
2
that
isfunction
the set in
of one
solutions
of yf(x)
= x=
.x
We’ll
call this
conic S.
The
graph
of
the
variable
2
isa called
a parabola.
=
x2
ph of the function in one variable f(x)
is called
parabola.
arabola
la
J2
Theinpoints
graph
in 1.)
theofThe
of f()
are
of the
f(x)).
shifted
right
and (x,
up
4: is, That is,
nts
the graph
are
points
ofpoints
(x,2form
f(x)).
f(x)parabola
the form
That
2
graph
einpoints
in the
). are
2
x
(x,
These
are the 2points
the points in the
Recall
that
the the
planar
moves
the graph
have
thehave
form
).transformation
2
x
These
theApoints
theR in
(x,form
(2,4) : Rin !
ane
coordinates
whose coordinates
second
of
are4. the
or an equation
plane right
2 and
That
is, A(2,4)
(x,coordinates,
y)first
= (xcoordinates,
+ 2,or
y + 4). Find
se second
their
firsttheir
are
the up
square
ofsqjiare
otherpoints
words,for
of the
xhere
(x, y)precompose
A(2,4)
(S).
Toform
this,
the equation for S, the equation y = x2 ,
ords,
ofpoints
the
form
(x,
y)doIiere
=
1
by A(2,4)
. The =inverse of
A(2,4) is A( 2, 4) and A( 2, 4) (x, y) = (x 2, y 4).
(Don’t
simplify
answer.)
recap.
of x
f(x)inyour
the
= x
is the
in theis plane
setset
as the set
graph
ofgraph
p,Tothe
the same
f(x) =
2
the2plane
setsame
as the
of the quadratic
in two variables
by the equation
of the quadratic
ssolutions
given bygiven
equationequation
in two variables
the equation
=
x2.
J
)(s)
2
A(
a
160
150
150
nts in the graph of f() are points of the form (x, f(x)). That is,
in the graph have the form (x, x
). These are the points in the
2
se second coordinates are the sqjiare of their first coordinates, or
ords, points of 2.)
theThe
formparabola
(x, y) xhere
shifted left 3 and down 2:
=
The planar transformation A( 3, 2) : R2 ! R2 moves points in the plane left
is, A(is 3,the2) (x,
y) =
3, yset 2). Find an equation
. the graph 3ofand
f(x)down
the plane
= x
22.in That
same
set (xas the
1
for A( equation
Totwo
do variables
this, precompose
s of the quadratic
3, 2) (S). in
given by the
the equation
equation for S by A( 3, 2) . The
inverse of A( 3, 2) is A(3,2) . (Don’t simplify your answer.)
150
o recap, the graph of f(x) = x
2 in the plane is the same set as the set
olutions of the quadratic equation in two variables given by the equation
y=x
9
3.) The parabola flipped over the x-axis:
150
The planar transformation
M : R2 ! R2 where
✓
◆
1 0
M=
0 1
he points in the graph of f(x) are points of the form (x, f()). That is,
points in the graph have the form (x, x
). These are the points in the
2
e whose second coordinates are the square of their first coordinates, or
ther words, points of the form (x, y) ihere
flips the plane over the x-axis. Written using row vectors, M (x, y) = (x, y).
Find an equation for M (S). To do this, precompose the equation for S by
M 1 . (Remember that M 1 = M , and don’t simplify your answer.)
(s)W
he graph of the function in one variable f(x)
rabola
J
161
=
x2
is called a parabola.
_____
abola
graph of the function
e points in the graph of f(x) are points of the form (x, f(x)). That is,
). These are the points in the
2
oints in the graph have the form (x, x
whose second coordinates are the square of their first coordinates, or
her words, points of the form (x, y) where
2 in the plane is the same set as the set
recap, the graph of f(x) = x
utions of the quadratic equation in two variables given by the equation
___
ill
J(x=x2
one variable f(x)
2
y=x
4.) The parabola flipped over the x = y line:
The planar transformation N : R2 ! R2 where
✓
◆
0 1
a
N=
1 0
h of the function in one variable f(x) = x
2 is called a parabola.
flips the plane over the x = y line. Written using row vectors, N (x, y) =
(y, x). Find an equation for N (S). (Remember that N 1 = N , and don’t
simplify your answer.)
=
2 is called a parabola.
x
I
5.) The parabola scaled by 2 in the y-coordinate:
ts in the graph of f(x) are points of the form2 (x, f(x)).
That is,
The planar transformation D : R ! R2 where
in the graph have the form (x, x
). These are the
2
✓ points
◆ in the
1
0
e second coordinates are the square of theirDfirst
= coordinates, or
0 2
ords, points of the form (x, y) where
=
is a diagonal matrix
that scales the y-coordinates of vectors by 2. That is,
D(x, y) =
(x, 2y). Find an equation for D(S). (Don’t simplify your answer,
= x2
, the graph and
of f(x)
the plane is the same set as the set
rememberinthat
✓
◆
s of the quadratic equation in two variables given
by1the
0 equation
1
D =
0 12
J
162
150
For #6-9, match the numbered function with its lettered graph.
6.) f (x) =
(
7.) g(x) =
(
x
1
2
1
x2
if x 0; and
if x < 0.
if x 0; and
if x < 0.
9.) m(x) =
(
x2
1
if x > 0; and
if x  0.
1
x2
if x > 0; and
if x  0.
C.)
‘I
‘I
I
‘I
I
I
A.)
8.) h(x) =
(
B.)
D.)
163