Ellipses and Hyperbolas - Mathwortman/1060text-eah.pdf · It’s an example of an ellipse. 204 Ellipses and Hyperbolas ... We’ll call this hyperbola H1, the unit hyperbola. 207

Ellipses and Hyperbolas

In this chapter we’ll see three more examples of conics.

EllipsesIf you begin with the unit circle, C1, and you scale x-coordinates by some

nonzero number a, and you scale y-coordinates by some nonzero number b,the resulting shape in the plane is called an ellipse.

Let’s begin again with the unit circle C1, the circle of radius 1 centered at(0, 0). We learned earlier that C1 is the set of solutions of the equation

x2 + y2 = 1

The matrix

(a 00 b

)scales the x-coordinates in the plane by a, and it scales

the y-coordinates by b. What’s drawn below on the right is

(a 00 b

)(C1),

which is the shape resulting from scaling C1 horizontally by a and verticallyby b. It’s an example of an ellipse.

204

Ellipses and HyperbolasIn this chapter we’ll revisit some examples of conies.

EllipsesIf you begin with the unit circle. C’. and you scale r-coordmates by some

nonzero number a. and you scale j-coordinates by some nonzero number b,the resulting shape in the plane is called an ellipse.

IoILet’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 equation9 2x-+y =1

The matrix D= ( ) scales the x-coordinates in the plane by a, and

it scales the y-coordinates by b. What’s drawn below is D(C’), which is theshape resulting from scaling C’ horizontally by a and vertically by b. It’s anexample of an ellipse.

H’

0Using POTS, the equation for this distorted circle, the ellipse D(C’), is

obtained by precomposing the equation for C’, the equation x2 + y2 = 1, bythe matrix

D1=(j ?)187

+II

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+0

+ II

Using POTS, the equation for this distorted circle, the ellipse

(a 00 b

)(C1),

is obtained by precomposing the equation for C1, the equation x2 + y2 = 1,by the matrix (

a 00 b

)−1=

(1a 00 1

b

)

Because

(1a 00 1

b

)replaces x with x

a and y with yb , the equation for the ellipse(

a 00 b

)(C1) is (x

a

)2+(yb

)2= 1

which is equivalent to the equation

x2

a2+y2

b2= 1

The equation for an ellipse described above is important. It’s repeated atthe top of the next page.

205








H’



D1=(j ?)187

+II

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+0

+ II

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+ II

+II

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Suppose a, b ∈ R− {0}. The equation for the ellipseobtained by scaling the unit circle

by a in the x-coordinate and by b in the y-coordinate is

x2

a2+y2

b2= 1

Example.

• The equation for the ellipse shown below is x2

25 + y2

4 = 1. In thisexample, we are using the formula from the top of the page with a = 5 andb = 2.

* * * * * * * * * * * * *

206








H’



D1=(j ?)187

tn

Circles are ellipsesA circle is a perfectly round ellipse.If we scale the unit circle by the same number r > 0 in both coordinates,

then we’ll stretch the unit circle evenly in all directions.

The result will be another circle, one whose equation is

x2

r2+y2

r2= 1

We can multiply both sides of this equation by r2 to obtain the equivalentequation

x2 + y2 = r2

which we had seen earlier as the equation for Cr, the circle of radius r centeredat (0, 0).

* * * * * * * * * * * * *

Hyperbolas from scalingWe’ve seen one example of a hyperbola, namely the set of solutions of the

equation xy = 1. We’ll call this hyperbola H1, the unit hyperbola.

207


EllipsesIf you begin with the unit circle. C’, and ou scale i-coordinates by some


b

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

9 9

The matrix D= ( ) scal:s th:x-coordinates in the plane by a, and


Using POTS. the equation for this distorted circle, the ellipse D(C’). isobtained by precomposing the equation for C1, the equation i2 + y2 = 1, bythe matrix

D’=(E5 )187

C,

r

C’


EllipsesIf you begin with the unit circle. C’, and ou scale i-coordinates by some


b

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

9 9

The matrix D= ( ) scal:s th:x-coordinates in the plane by a, and


Using POTS. the equation for this distorted circle, the ellipse D(C’). isobtained by precomposing the equation for C1, the equation i2 + y2 = 1, bythe matrix

D’=(E5 )187

C,

r

C’








H’



D1=(j ?)187

Suppose c > 0. We can horizontally stretch the unit hyperbola H1 using

the diagonal matrix

(c 00 1

). This is the matrix that scales the x-coordinate

by c and does not alter the y-coordinate. The resulting shape in the plane,(c 00 1

)(H1), is also called a hyperbola.

The equation for the hyperbola

(c 00 1

)(H1) is obtained by precomposing

the equation for H1, the equation xy = 1, with

(c 00 1

)−1=

(1c 00 1

). This is

the matrix that replaces x with xc and does not alter y. Thus, the equation

for the hyperbola

(c 00 1

)(H1) is(x

c

)y = 1

which is equivalent toxy = c

From now on, we’ll call this hyperbola Hc.

To summarize:208

.xI—cx(c,i)

D(H)








H’



D1=(j ?)187

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+ II+II

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Jo

=)1c\fjo‘4O,)ço)

Let c > 0. Then Hc is the hyperbolathat is the set of solutions of the equation xy = c.

Example.

• The equation for the hyperbola H2, obtained by scaling the unithyperbola by 2 in the x-coordinate is xy = 2.

Hyperbolas from flipping

We can flip the hyperbola Hc over the y-axis using the matrix

(−1 00 1

),

the matrix that replaces x with −x and does not alter y.

209

Ellipses and HyperbolasIn this chapter we’ll revisit some examples of conics.

EllipsesIf you begin with the unit circle, C’, and you scale x-coordinates by some


D(c’)C)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 equation9 9

The matrix D= ( ) scai:s the -coordinates in the plane by a, and

it scales the g-coordinates by b. What’s drawn below is D(C’), v.hich is theshape resulting from scaling C’ horizontally by a and vertically by b. It’s anexample of an ellipse.

Using POTS. the equation for this distorted circle, the ellipse D(C’). isobtained by precomposing the equation for C’, the equation x2 + y2 = 1, bythe matrix

D1=( ?)187

I1c

NEllipses and Hyperbolas

In this chapter we’ll revisit some examples of conics.

EllipsesIf you begin with the unit circle, C’, and you scale x-coordinates by some


D(c’)C)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 equation9 9

The matrix D= ( ) scai:s the -coordinates in the plane by a, and

it scales the g-coordinates by b. What’s drawn below is D(C’), v.hich is theshape resulting from scaling C’ horizontally by a and vertically by b. It’s anexample of an ellipse.

Using POTS. the equation for this distorted circle, the ellipse D(C’). isobtained by precomposing the equation for C’, the equation x2 + y2 = 1, bythe matrix

D1=( ?)187

I1c


EllipsesIf you begin with the unit circle, C’, and you scale i-coordinates by some


C)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+ = 1

The matrix D= ( ) scales the i-coordinates in the plane by a, and


—I

Using POTS, the equation for this distorted circle, the ellipse D(C’), isobtained by precomposing the equation for C’, the equation x2 + y2 = 1, bythe matrix

D1=(0)

187

x

TT1-

,J,.J,

%____;

c0

--‘

I’II 0

0_

-0

x

TT1-

,J,.J,

%____;

c0

--‘

I’II 0

0_

-0

x

TT1-

,J,.J,

%____;

c0

--‘

I’II 0

0_

-0

The shape resulting from flipping the hyperbola Hc over the y-axis is alsocalled a hyperbola. Its equation is obtained by precomposing the equation

for Hc, the equation xy = c, with the inverse of

(−1 00 1

), which is

(−1 00 1

)itself, the matrix that replaces x with −x.

So the equation for Hc after being flipped over the y-axis is

(−x)y = c

which is equivalent toxy = −c

To summarize:

Let c > 0. The equation for Hc flippedover the y-axis is xy = −c.

210

tr’

x

TT1-

,J,.J,

%____;

c0

--‘

I’II 0

0_

-0

Example.

• The equation for H3 after being flipped over the y-axis is xy = −3.

* * * * * * * * * * * * *

Ellipses and hyperbolas are conicsThe equations that we’ve seen for ellipses and hyperbolas in this chapter

are quadratic equations in two variables: x2

a2 + y2

b2 = 1, xy = c, and xy = −c.Thus, the solutions of these equations, the ellipses and hyperbolas above, areexamples of conics.

211

Exercises

For #1-6, match the numbered pictures with the lettered equations.

1.) 4.)

2.) 5.)

3.) 6.)

A.) xy = 4 B.) x2 + 4y2 = 1 C.) x2 + y2 = 9

D.) xy = −4 E.) x2

9 + y2

16 = 1 F.) x2

25 + y2 = 1

212

(J Ellipses and HyperbolasIn this chapter we’ll revisit some examples of comcs.



E1

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

9 9

The matrix D ( ) scales th: i-coordinates in the plane by a, and

it scales the y-coordinates by b. What’s drawn below is D(C’). which is theshape resulting from scaling C’ horizont ally by a and vertically by b. It’s anexample of an ellipse.

Using POTS, the equation for this distorted circle, the ellipse D(C1), isobtained by precomposing the equation for C’, the equation x2 + y2 = 1, bythe matrix

D1=( ?)187

3

-3

Ellipses and HyperbolasIn this chapter we’ll revisit some examples of comcs.



E1


9 9




D1=( ?)187

3

-3

Ellipses and HyperbolasIn this chapter we’ll revisit some examples of comcs.



E1


9 9




D1=( ?)187

3

-3

For #7-10, write an equation for each of the given shapes in the plane. Youranswers should have the form (y − q) = a(x − p), (x − p)2 + (y − q)2 = r2,

(x − p)(y − q) = c, or (x−p)2a2 + (y−q)2

b2 = 1. (As with the x- and y-axes, thedotted lines are not part of the shapes. They’re just drawn to provide a pointof reference.)

7.) 9.)

8.) 10.)

For #11-14, multiply the matrices.

11.)

(1 −1−1 1

)(0 32 −3

)13.)

(1 00 1

)(5 78 9

)

12.)

(2 00 3

)(12 00 1

3

)14.)

(3 −21 0

)(2 10 1

)213

A For #7-10 write an equation for each of the given shapes in the plane.dotted lines are not part of the shapes. theyre just drawn to provide a

point of reference.)

8.) 10.)

11.) 1) (19) (2 0 ( 0-. 0 3)\O

3—3 13.) (a ) (

“3 —914.) 0

79

(2 11

7.)(f)

9.)

5

(35:


194



8.) 10.)

11.) 1) (19) (2 0 ( 0-. 0 3)\O

3—3 13.) (a ) (

“3 —914.) 0

79

(2 11

7.)(f)

9.)

5

(35:


194



8.) 10.)

11.) 1) (19) (2 0 ( 0-. 0 3)\O

3—3 13.) (a ) (

“3 —914.) 0

79

(2 11

7.)(f)

9.)

5

(35:


194



8.) 10.)

11.) 1) (19) (2 0 ( 0-. 0 3)\O

3—3 13.) (a ) (

“3 —914.) 0

79

(2 11

7.)(f)

9.)

5

(35:


194

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Ellipses and Hyperbolas - Mathwortman/1060text-eah.pdf · It’s an example of an ellipse. 204 Ellipses and Hyperbolas ... We’ll call this hyperbola H1, the unit hyperbola. 207