8.6 Translate and Classify Conic Sections

What is the general 2nd degree equation for any conic?

What information can the discriminant tell you about a conic?

The equation of any conic can be written

in the form-

Called a general 2nd degree equation

2 2 0Ax Bxy Cy Dx Ey F

Identify the line(s) of symmetry for each conic section in Examples 1 – 4.

SOLUTION

For the circle in Example 1, any line through the center (2, – 3) is a line of symmetry.

For the hyperbola in Example 2 x = – 1 and y = 3 are lines of symmetry

For the parabola in Example 3, y = 3 is a line of symmetry.

For the ellipse in Example 4, x = 4 and y = 2 are lines of symmetry.

Identify the line(s) of symmetry for each conic section in Examples 1 – 4.

SOLUTION

7. (x – 5)2

64 + (y)2

16 = 1

Identify the line(s) of symmetry for the conic section.

For the ellipse the lines of symmetry are x = 5 and y = 0.

ANSWER

8. (x + 5)2 = 8(y – 2).

For parabola the lie of symmetry are x = – 5

ANSWER

Identify the line(s) of symmetry for the conic section.

9. (x – 1)2

49 –(y – 2)2

121 = 1

For horizontal lines of symmetry are x = 1 and y = 2.

ANSWER

Identify the line(s) of symmetry for the conic section.

Circles

Can be multiplied out to look like this….

2 2( 1) ( 2) 16x y

2 2 2 4 11 0x y x y

Ellipse

Can be written like this…..

22( 1)

( 1) 14

xy

2 24 2 8 1 0x y x y

Parabola

Can be written like this…..

2( 6) 4( 8)y x

2 12 4 4 0y y x

Hyperbola

Can be written like this…..

22 ( 4)

( 4) 19

yx

2 29 72 8 1 0x y x y

How do you know which conic it is when it’s been multiplied

out?

• Pay close attention to whose squared and whose not…

• Look at the coefficients in front of the squared terms and their signs.

Circle Both x and y are

squared

And their coefficients are the same number and sign

2 2 2 4 11 0x y x y

Ellipse• Both x and y are

squared• Their coefficients are

different but their signs remain the same.

2 24 2 8 1 0x y x y

Parabola• Either x or y is

squared but not both

2 12 4 4 0y y x

Hyperbola

• Both x and y are squared

• Their coefficients are different and so are their signs.

2 29 72 8 1 0x y x y

You Try!

0343.10

036164.9

0782.8

0593033.7

0164y8x2x6.

046y6x2y2x.5

0314y12x2y2x4.

03023y25x3.

041y20x22x2.

032x24y2x1.

22

22

22

22

yyx

yxyx

xyx

xyx

1.Ellipse

2.Parabola

3.Hyperbola

4.Circle

5.Hyperbola

6.Parabola

7.Circle

8.Ellipse

9.Hyperbola

10.Ellipse

When you want to be sure…

of a conic equation, then find the type of conic using discriminate information:

Ax2 +Bxy +Cy2 +Dx +Ey +F = 0

B2 − 4AC < 0, B = 0 & A = C Circle

B2 − 4AC < 0 & either B≠0 or A≠C Ellipse

B2 − 4AC = 0 Parabola

B2 − 4AC > 0 Hyperbola

Classify the Conic

2x2 + y2 −4x − 4 = 0

Ax2 +Bxy +Cy2 +Dx +Ey +F = 0

A = 2

B = 0

C = 1

B2 − 4AC = 02 − 4(2)(1) = −8

B2 − 4AC < 0, the conic is an ellipse

Write the equation in standard form by completing the square

01824 22 yxyx

______1___)2(4___2 22 yyxx

1)2(42 22 yyxx

)1)(4(11)12(4)12( 22 yyxx

4)1(4)1(22 yx

4

4

4

)1(4

4

)1( 22

yx

11

)1(

4

)1( 22

yx

12

22

Steps to Complete the Square1. Group x’s and y’s. (Boys with the boys and

girls with the girls) Send constant numbers to the other side of the equal sign.

2. The coefficient of the x2 and y2 must be 1. If not, factor out.

3. Take the number before the x, divide by 2 and square. Do the same with the number before y.

4. Add these numbers to both sides of the equation. *(Multiply it by the common factor in #2)

5. Factor

An ellipse is defined by the equation 4x2 + 9y2 – 16x + 18y = 11. Write the standard equation and identify the coordinates of the center, vertices, co-vertices, and foci. Sketch the graph of the ellipse.

4(x2 – 4x) + 9(y2 + 2y) = 11

4x2 – 16x + 9y2 + 18y = 11

4(x2 – 4x + 4) + 9(y2 + 2y + 1) = 11 + 4(4) + 9(1)4(x – 2)2 + 9(y + 1)2 = 36

22 9(y 1)4(x 2) 3636 36 36

22 (y 1)(x 2)1

9 4

Graph the Conic2x2 + y2 −4x − 4 = 0

2x2 −4x + y2 = 4

2(x2 −2x +___)+ y2 = 4 + ___ (−2/2)2= 1

2(x2 −2x +1)+ y2 = 4 + 2(1)

2(x−1)2 + y2 = 6

V(1±√6), CV(1±√3)

166

)1(2 22

yx

163

)1( 22

yx

Complete the Square

10. Classify the conic given by x2 + y2 – 2x + 4y + 1 = 0. Then graph the equation.

Note that A = 1, B = 0, and C = 1, so the value of the discriminant is: B2 – 4AC = 02 – 4(1)(1) = – 4

SOLUTION

Because B2 – 4AC < 0 and A = C, the conic is an circle.To graph the circle, first complete the square in both x and y simultaneity .

x2 + y2 – 2x + 4y + 1 = 0

x2 – 2x +1+ y2 + 4y + 4 = 4

(x – 1)2 +( y + 2)2 = 4

From the equation, you can see that (h, k) = (– 1, 2), r = 2 Use these facts to draw the circle.

ANSWER

Physical Science

In a lab experiment, you record images of a steel ball rolling past a magnet. The equation 16x2 – 9y2 – 96x + 36y – 36 = 0 models the ball’s path.• What is the shape of the path ?

• Write an equation for the path in standard form.

• Graph the equation of the path.

SOLUTION

STEP 1 Identify the shape. The equation is a general second-degree equation with A = 16, B = 0, and C = – 9. Find the value of the discriminant.B2 – 4AC = 02 – 4(16)(– 9) = 576

Because B2 – 4AC > 0, the shape of the path is a hyperbola.

STEP 2 Write an equation. To write an equation of the hyperbola, complete the square in both x and y simultaneously.

16x2 – 9y2 – 96x + 36y – 36 = 0(16x2 – 96x) – (9y2 – 36y) = 36

16(x2 – 6x + ? ) – 9(y2 – 4y + ? ) = 36 + 16( ? ) – 9( ? )

16(x2 – 6x + 9) – 9(y2 – 4y + 4) = 36 + 16(9) – 9(4)16(x – 3)2 – 9(y – 2)2 = 144

(x – 3)2

9–

(y –2)2 16

= 1

STEP 3 Graph the equation. From the equation, the transverse axis is horizontal, (h, k) = (3, 2),

a = 9 = 3 and b = 16. = 4

The vertices are at (3 + a, 2), or (6, 2) and (0, 2).

See page 530

What is the general 2nd degree equation for any conic?

What information can the discriminant tell you about a conic?

B2- 4AC < 0, B = 0, A = CCircle

B2- 4AC < 0, B ≠ 0, A ≠ CEllipse

B2- 4AC = 0, Parabola

B2- 4AC > 0 Hyperbola

2 2 0Ax Bxy Cy Dx Ey F

8.6 Assignment

Page 531, 23-43 odd