In thisInchapter
we’ll we’ll
see three
examples of conics.
this chapter
revisitmore
some examples of conies.
Ellipses
Ellipses
‘I
I
IoI
I-o
I-o
on
on
-
If you Ifbegin
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andand
youyouscale
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you begin
the unit
circle.
r-coordmatesby
scale
by some
some
nonzero
number
a, and
scalescale
y-coordinates
by
some
nonzero
number
nonzero
number
a. you
and you
j-coordinates by some nonzero number b,
b,
the resulting
shapeshape
in the
is called
an an
ellipse.
the resulting
in plane
the plane
is called
ellipse.
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I
II
Let’s begin again with the unit circle C’, the circle of radius 1 centered at
II
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the unit
C 1 , set
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learned
0). We
that1 circle
C’ is the
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of the 1equation
(0, 0). We learned earlier that C is the
set
9
2 of solutions of the equation
x-+y =1
The matrix
D
()
x2 + y 2 = 1
scales the x-coordinates in the plane by a, and
a 0 =
The matrix
scales the x-coordinates in the plane by a, and it scales
0 y-coordinate
b
it scales the
s by b. What’s drawn below is D(C’), which
is the
shape resulting from scaling C’ horizontally by a and vertically a
by 0b. It’s 1an
(C ),
the y-coordinates
by
b.
What’s
drawn
below
on
the
right
is
example of an ellipse.
0 b
which is the shape resulting from scaling C 1 horizontally
by a and vertically
H’
by b. It’s an example of an ellipse.
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op
0
204
0
+
II
II
+
+
-co
-co
0
187
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+
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D1=(j
4
4
+
op
op
z:3
z:3
Using POTS, the equation for this distorted circle, the ellipse D(C’), is
obtained by precomposing the equation for C’, the equation x
2+y
2 = 1, by
the matrix
+
_
Ellipses
and
Ellipses
andHyperbolas
Hyperbolas
I-o
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number b,
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44
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nonzero number a. and you scale j-coordinates by some
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a 0
Using POTS, the Ellipses
equation for this
distorted
circle, the ellipse
(C 1 ),
and
Hyperbolas
0 b
1
is obtained
by chapter
precomposing
the equation
for Cof, conies.
the equation x2 + y 2 = 1,
In this
we’ll revisit
some examples
by the matrix
−1 1 Ellipses
a 0
0
=C’. aand1 you scale r-coordmates by some
If you begin with the unit
circle.
0 b
0 b
++ 00
-co
II
scales the x-coordinates in the plane by a, and
+
()
9
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=
4
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The matrix D
x-+y 2 =1
z:3
op
Let’s begin again with the unit circle C’, the circle of radius 1 centered at
(0, 0). We learned earlier that C’ is the set of solutions of the equation
+
0
it scales the y-coordinates by b. What’s drawn below is D(C’), which is the
shape resulting from scaling C’ horizontally by a and vertically by b. It’s an
example of an ellipse.
H’
1
Because a
0
a 0
(C 1 ) is
0 b
0
0
1
b
replaces x with
x
a
x 2
and y with yb , the equation for the ellipse
y 2
+
=1
a
b
Using POTS, the equation for this distorted circle, the ellipse D(C’), is
which obtained
is equivalent to the equation
by precomposing the equation for C’, the equation x
2+y
2 = 1, by
2
2
y
x
the matrix
+ 2 =1
a2 D1=(j
b
?)
187
The equation for an ellipse described above is important. It’s repeated at
the top of the next page.
205
Ellipses and Hyperbolas
a, b ∈ R − {0}. The equation for the ellipse
n this chapter we’ll revisit Suppose
some examples of conies.
obtained by scaling the unit circle
lipses
by a in the x-coordinate and by b in the y-coordinate is
you begin with the unit circle. C’. and you scale
by some
x2 r-coordmates
y2
+
=
1
zero number a. and you scale j-coordinates by some
nonzero
number
b,
a2 b2
resulting shape in the plane is called an ellipse.
IoI
et’s begin again with the unit circle C’, the circle of radius 1 centered at
0). We learned earlier that C’ is the set of solutions of the equation
x-+y 2 =1
9
Example.
he matrix D
()
scales the x-coordinates in the plane by a, and
2
2
• The equation for the ellipse shown below is x25 + y4 = 1. In this
cales the y-coordinates by b. What’s drawn below is D(C’), which is the
example, we are using the formula from the top of the page with a = 5 and
pe resulting from scaling C’ horizontally by a and vertically by b. It’s an
b = 2.
=
mple of an ellipse.
H’
tn
0
sing POTS, the equation for this distorted circle, the ellipse D(C’), is
ained by precomposing the equation for C’, the equation x
2+y
2 = 1, by
matrix
*
*
*
*
*
*
*
*
*
*
*
D1=(j
?)
187
206
*
*
Let’s begin again with the unit circle
the circle of radius 1 centered at
(0. 0). We learned earlier that C’ is the set of solutions of the equation
Ellipses and Hyp
9
Circles are ellipses
The matrix D
()
9
In this chapter we’ll
revisit
some examples o
scal:s th:x-coordinates in the plane by a, and
Ellipses
A circle is a perfectly
round ellipse.
=
with which
the unit
circle. C’, and o
it scales
by the
b. What’s
drawnIfbelow
D(C’),
is the
If we
scale the
they-coordinates
unit circle by
same number
ryou
> begin
0is in
both
coordinates,
a, by
andb. you
from scaling C’ horizontally nonzero
resultingthe
by a andnumber
vertically
It’s scale
an y-coordinate
then shape
we’ll stretch
unit circle evenly in all directions.
the resulting shape in the plane is called an el
example of an ellipse.
C,
C’
Ellipses and Hyperbolas
r
this chapter we’ll revisit some examples of conies.
ipses
you begin with the unit circle. C’. and you scale r-coordmates by some
begin
the unit
circle
th
Using POTS. the equation for this distortedLet’s
circle,
ellipsewith
the again
D(C’).
is
zero number a. The
and result
you scale
j-coordinates
nonzero
number
by some
b,
will
be
another
circle,
one
whose
equation
is
learned
(0. 0).
obtained by precomposing the equation for C
the We
,
1
equation
2 earlier
i
1, byC’ is the set of
+y
2 =that
resulting shape in the plane is called an ellipse. 2
2
9
9
the matrix
y
x
)
+ 2 =1
2
rD’=(E5
r
The2 matrix D
scal:s th:x-coord
We can multiply both sides of this equation
by
r
to
obtain
the
equivalent
=
187
it scales the y-coordinates by b. What’s drawn
equation
2
2
2 shape resulting from scaling C’ horizontally b
x +y =r
of an
which we had seen earlier as the equation for C rexample
, the circle
of ellipse.
radius r centered
at (0, 0).
IoI
()
C’
et’s begin again with the unit circle C’, the circle of radius 1 centered at
). We learned earlier that C’ is the set of solutions of the equation
*
*
*
*
*
*
*
*
*
*
*
9
2
x-+y =1
he matrix D
()
Hyperbolas
from
scaling
scales the
x-coordinate
s in
=
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the plane by a, and
We’ves seen
example
a hyperbola,
ales the y-coordinate
by b. one
What’s
drawnofbelow
is D(C’), namely
which
isthe
the set of solutions of the
1 Using POTS. the equation for this distorte
equation
1. We’ll call
hyperbola
,b.the
pe resulting from
scalingxyC’=horizontally
by this
vertically H
a and
It’sunit
by obtained
anbyhyperbola.
precomposing the equation for C
,
1
mple of an ellipse.
the matrix
H’
0
D’=(E5
187
sing POTS, the equation for this distorted circle, the
207ellipse D(C’), is
ined by precomposing the equation for C’, the equation x
2+y
2 = 1, by
matrix
D1=(j
?)
)
unit circle C’, the circle of radius 1 centered at
1
> the
0. equation
We
C’ is the set ofSuppose
solutionsc of
canhorizontally stretch the unit hyperbola H using
c 0
. This is the matrix that scales the x-coordinate
0 1
by
alter
y-coordinate. The resulting shape in the plane,
cales the x-coordinate
plane
in thenot
by the
a, and
c and
s does
c 0
(H 1 ), is also called a hyperbola.
b. What’s drawn
0 1below is D(C’), which is the
x-+y 2 =1the diagonal matrix
9
-
H’
-
-
C’ horizontally by a and vertically by b. It’s an
(c,i)
on
on
.xI—cx
I
II
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I
I
fj
c\
o
II
‘I
=)
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1
)
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,)ço
D1=(j
‘4O
for this distorted circle, the ellipse D(C’), is
e equation for C’, the equation x
2+y
2 = 1, by
I-o
I-o
D(H)
187
Jo
c 0
The equation for the hyperbola
(H 1 ) is obtained by precomposing
0 1
−1 1 c
0
0
. This is
the equation for H 1 , the equation xy = 1, with
= c
0 1
0 1
x
the matrix that replaces
x with c and does not alter y. Thus, the equation
c 0
for the hyperbola
(H 1 ) is
0 1
x
y=1
c
which is equivalent to
xy = c
From now on, we’ll call this hyperbola H c .
II
II
+
+
-co
-co
+
0
+
+
208
0
4
To summarize:
4
II
z:3
II
_
op
op
z:3
+
+
_
op
op
et’s begin again with the unit circle C’, the circle of radius 1 centered at
0). We learned earlier that C’ is the set of solutions of the equation
9
he matrix D
=
()
9
scai:s the -co ordinates in the plane by a, and
c
Letdrawn
c > 0.below
Then
is the
hyperbola
cales the g-coordinates by b. What’s
is H
D(C’),
v.hich
is the
is the setbyofa solutions
of
the
equation
pe resulting from scaling C’that
horizontally
and vertically by b. It’s an xy = c.
mple of an ellipse.
I1c
%____;
TT1,J,.J,
TT1,J,.J,
TT1,J,.J,
%____;
pses and Hyperbolas
Ellipses and Hyperbolas
%____;
l revisit some examples In
of conics.
this chapter we’ll revisit some examples of conies.
D(c’)
187
II
0
I’
N
C)
0
II
II--‘
0
--‘
c0
c0
I’
I’
c0
?)
hyperbola by 2 in the x-coordinate is xy = 2.
--‘
xx
x
sing POTS. the equation
for this distorted circle, the ellipse D(C’). is
Ellipses
ained
precomposing
by
the
equation
for C’, thebyequation
2+y
x
2 = 1, by
he unit circle, C’, and you
x-coordinates
scale
some
begin
with the unit
circle, C’, and you scale i-coordinates by some
Example.If you
matrix
nd you scale y-coordinates
by number
some nonzero
b, y-coordinates by some nonzero number b,
a, andnumber
nonzero
you scale
the plane is called an
ellipse.
D1=(
• the
The
equation
for inthe
H 2 , an
obtained
shape
resulting
thehyperbola
plane is called
ellipse. by scaling the unit
ith the unit circle C’, the
circle
of radius
1 centered
at circle
Let’s
begin
again with
the unit
the circle of radius 1 centered at
lier that C’ is the set (0,
of
of 0).
solutions
the
equation
We learned earlier that C’ is the set of solutions of the equation
()
9
9
+
Hyperbolas from flipping
=
1
()
0_
-0
I1c
-0
0_
example of an ellipse.
0_
-0
scai:s the -co ordinates
in the
byc a, scales
and
D plane H
i-coordinates
the plane−1
by 0a, and
We can The
flip matrix
the hyperbola
over thethey-axis
using theinmatrix
,
=
0
1
nates by b. What’s drawn
below
is y-coordinates
D(C’), v.hich by
is the
it scales
the
b.and
What’s
below
the
matrix
that
replaces
x
with
−x
does drawn
not alter
y. is D(C’), which is the
scaling C’ horizontallyshape
by a resulting
and vertically
by
b.
It’s
an
from scaling C’ horizontally by a and vertically by b. It’s an
—I
209
quation for this distorted
circle,
the the
ellipse
D(C’).for
is this distorted circle, the ellipse D(C’), is
POTS,
Using
equation
osing the equation for obtained
C’, the equation
2+y
x
1, by
2 =the
by precomposing
equation for C’, the equation x
2+y
2 = 1, by
the matrix
0)
D1=(
?)
D1=(
0
II
I’
--‘
c0
,J,.J,
x
TT1-
%____;
The shape resulting from flipping the hyperbola H c over the y-axis is also
called a hyperbola. Its equation is obtained byprecomposing
the equation
−1
0
−1
0
for H c , the equation xy = c, with the inverse of
, which is
0 1
0 1
itself, the matrix that replaces x with −x.
So the equation for H c after being flipped over the y-axis is
(−x)y = c
which is equivalent to
xy = −c
To summarize:
0_
tr’
210
-0
Let c > 0. The equation for H c flipped
over the y-axis is xy = −c.
Example.
• The equation for H 3 after being flipped over the y-axis is xy = −3.
*
*
*
*
*
*
*
*
*
*
*
*
*
Ellipses and hyperbolas are conics
The equations that we’ve seen for ellipses and hyperbolas in this chapter
2
2
are quadratic equations in two variables: xa2 + yb2 = 1, xy = c, and xy = −c.
Thus, the solutions of these equations, the ellipses and hyperbolas above, are
examples of conics.
211
ses and Hyperbolas
Exercises
For #1-6,
match the numbered pictures with the lettered equations.
visit some examples
of comcs.
1.)
4.)
(J
unit circle, C’, and you scale i-coordinates by some
you scale y-coordinates by some nonzero number b,
e plane is called an ellipse.
this chapter
examples
of comcs.
In thisIn chapter
we’ll we’ll
revisitrevisit
some some
examples
of comcs.
Ellipses
Hyperbolas
Ellipses
andand
Hyperbolas
Ellipses
Ellipses
you begin
withunit
the circle,
unit circle,
C’, you
andscale
i-coordinates
you scale
If youIfbegin
by some
with the
C’, and
i-coordinates
by some
number
nonzero
and scale
you scale
y-coordinates
nonzero
by some
number
number
nonzero
b,
a, anda, you
y-coordinates
nonzero
by some
number
b,
E1
the resulting
the plane
is called
an ellipse.
the resulting
shapeshape
in theinplane
is called
an ellipse.
2.)
3
5.)
3
the unit circle C’, the circle of radius 1 centered at
that C’ is the set of solutions of the equation
9
9
E1
E1
scales th: i-coordinates in the plane by a, and
-3 -3
s by b. What’s drawn Let’s
belowbegin
is D(C’). which
is the
the
unit circle
withunit
the circle
C’,circle
of radius
1 centered
Let’s begin again again
at
with the
circle
C’, the
of radius
1 centered
at
ing C’ horizont ally by
and
vertically
a
by
b.
It’s
an
We learned
0).learned
thatis C’
theofset
of solutions
the equation
(0, 0).(0,We
earlierearlier
that C’
theisset
solutions
of theofequation
)
3.)
9
99
6.) 9
( )( )
D
The matrix
D
The matrix
th: i-coordinates
the plane
scalesscales
th: i-coordinates
in theinplane
by a, by
anda, and
it scales
the y-coordinates
b. What’s
is D(C’).
it scales
the y-coordinates
by b. by
What’s
drawndrawn
belowbelow
is D(C’).
whichwhich
is the is the
shape
resulting
scaling
C’ horizont
ally
by avertically
and vertically
shape resulting from from
scaling
C’ horizont
ally by
a and
by b. by
It’sb.anIt’s an
example
an
ellipse.
of
example of an ellipse.
ation for this distorted circle, the ellipse 1
D(C
)
, is
g the equation for C’, the equation x
2+y
2 = 1, by
?) xy = 4
D1=( A.)
187
B.) x2 + 4y 2 = 1
2
2
C.) x2 + y 2 = 9
2
y
x
2
D.)
xyUsing
= −4POTS,
E.)forx9 this
+for16
= 1distorted
F.) ellipse
+ y,
=is,
1 is
the equation
this
the
Using
)
1
D(C
POTS,
the equation
25D(C
distorted
circle,circle,
the ellipse
)
1
obtained
by precomposing
the equation
forthe
the equation
C’,equation
2=
x
obtained
= 1, by
by precomposing
the equation
for C’,
+y
2
2+y
x
1, by
2
212
the
matrix
the matrix
?) ?)
D1=(
D1=(
187
187
9.)
(f)
For #7-10, write an equation for each of the given shapes in the plane. Your
answers should have the form (y − q) = a(x − p), (x − p)2 + (y − q)2 = r2 ,
(y−q)2
(x−p)2
(xFor
− p)(y
−
q)
=
c,
or
+
=of1.the(As
with the x- and y-axes, the
2
write an equation
a2
beach
for
given shapes in the plane.
#7-10
write
For
an
each
of
equation
for
the
given shapes in the plane.
#7-10
dotted
lines lines
are not
of the
shapes.
They’re
just drawn
to
a point
are partpart
dotted
shapes.
of the
drawn
to provide
provide
theyre
dotted
lines are notnot
shapes.
part of the
justjust
drawn
to provide
a a
theyre
of
reference.)
point
of reference.)
A
point of reference.)
7.)
10.)
7.) 7.)
9.) 9.)
(f)
(f)
9.)
(35:
5 5
For #7-10 write an equation for each of the given s
dotted lines are not part of the shapes. theyre just
point of reference.)
A
8.)
8.) 8.)
10.)10.)10.)
7.)
9.)
(f)
(35:
(35:
e matrices.
(
3
—3
13.)
14.)
(a ) (
“3
5
7
9
(2 1
1
—9
0
8.)
194
10.)
multiply
matrices.
#11-14,
ForFor
multiply
thethe
matrices.
#11-14,
For
#11-14,
multiply
the matrices.
11.)
11.)11.)
(a (a) )( (
((
1)
3
1
−1
0
3
1)
3
−1 1 —32—3−3
1
13.)
13.)
13.)
0
((
0 10 (2
0
(2
02 0 0
19)19)
12.) 0 3)\O 2 1
0 3)\O
0 3
0 3
-.
07 7 5 7
91 9 8 9
—9
(21 1 2 1
“3 “3—9 3(2 −2
14.)14.) 14.) 0
0 1 01 1 0 1
-.
194
194
213
For #11-14, multiply the matrices.
11.)
1)
3
13.)