8.6 Translate and Classify Conic Sections

Transcription

1 8.6 Translate and Classify Conic Sections Where are the symmetric lines of conic sections? What is the general 2 nd degree equation for any conic? What information can the discriminant tell you about a conic?

2 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

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

4 Identify the line(s) of symmetry for the conic section. 7. (x 5) (y) 2 16 = 1 ANSWER For the ellipse the lines of symmetry are x = 5 and y = 0. Identify the line(s) of symmetry for the conic section. 8. (x + 5) 2 = 8(y 2). ANSWER For parabola the lie of symmetry are x = 5

5 Identify the line(s) of symmetry for the conic section. 9. (x 1) 2 49 (y 2)2 121 = 1 ANSWER For the hyperbola, the lines of symmetry are x = 1 and y = 2.

6 The equation of any conic can be written in the form- Called a general 2 nd degree equation

7 Circles Can be multiplied out to look like this.

8 Ellipse Can be written like this..

9 Parabola Can be written like this..

10 Hyperbola Can be written like this..

11 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.

12 Both x and y are squared And their coefficients are the same number and sign Circle

13 Both x and y are squared Their coefficients are different but their signs remain the same. Ellipse

14 Either x or y is squared but not both Parabola

15 Both x and y are squared Their coefficients are different and so are their signs. Hyperbola

16 You Try! 1. Ellipse 2. Parabola 3. Hyperbola 4. Circle 5. Hyperbola 6. Parabola 7. Circle 8. Ellipse 9. Hyperbola 10. Ellipse

17 When you want to be sure of a conic equation, then find the type of conic using discriminate information: Ax 2 +Bxy +Cy 2 +Dx +Ey +F = 0 B 2 4AC < 0, B = 0 & A = C Circle B 2 4AC < 0 & either B 0 or A C Ellipse B 2 4AC = 0 Parabola B 2 4AC > 0 Hyperbola

18 Classify the Conic 2x 2 + y 2 4x 4 = 0 Ax 2 +Bxy +Cy 2 +Dx +Ey +F = 0 A = 2 B = 0 C = 1 B 2 4AC = 0 2 4(2)(1) = 8 B 2 4AC < 0, the conic is an ellipse

19 Write the equation in standard form by completing the square

20 Steps to Complete the Square 1. 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 x 2 and y 2 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

21 Write the equation in standard form by completing the square

22 An ellipse is defined by the equation 4x 2 + 9y 2 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. 4x 2 16x + 9y y = 11 4(x 2 4x) + 9(y 2 + 2y) = 11 4(x 2 4x + 4) + 9(y 2 + 2y + 1) = (4) + 9(1) 4(x 2) 2 + 9(y + 1) 2 = 36

23 Graph the Conic 2x 2 + y 2 4x 4 = 0 2x 2 4x + y 2 = 4 Complete the Square 2(x 2 2x + )+ y 2 = 4 + ( 2/2) 2 = 1 2(x 2 2x +1)+ y 2 = 4 + 2(1) 2(x 1) 2 + y 2 = 6 V(1± 6), CV(1± 3)

24 10. Classify the conic given by x 2 + y 2 2x + 4y + 1 = 0. Then graph the equation. SOLUTION Note that A = 1, B = 0, and C = 1, so the value of the discriminant is: B 2 4AC = 0 2 4(1)(1) = 4 Because B 2 4AC < 0 and A = C, the conic is an circle. To graph the circle, first complete the square in both x and y simultaneity. x 2 + y 2 2x + 4y + 1 = 0 x 2 2x +1+ y 2 + 4y + 4 = 4 (x 1) 2 +( y + 2) 2 = 4 ANSWER From the equation, you can see that (h, k) = ( 1, 2), r = 2 Use these facts to draw the circle.

25 Physical Science In a lab experiment, you record images of a steel ball rolling past a magnet. The equation 16x 2 9y 2 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. B 2 4AC = 0 2 4(16)( 9) = 576 Because B 2 4AC > 0, the shape of the path is a hyperbola.

26 STEP 2 Write an equation. To write an equation of the hyperbola, complete the square in both x and y simultaneously. 16x 2 9y 2 96x + 36y 36 = 0 (16x 2 96x) (9y 2 36y) = 36 16(x 2 6x +? ) 9(y 2 4y +? ) = (? ) 9(? ) 16(x 2 6x + 9) 9(y 2 4y + 4) = (9) 9(4) 16(x 3) 2 9(y 2) 2 = 144 (x 3) 2 (y 2)2 = 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

27 What is the general 2 nd degree equation for any conic? What information can the discriminant tell you about a conic? B 2-4AC < 0, B = 0, A = C Circle B 2-4AC < 0, B 0, A C Ellipse B 2-4AC = 0, Parabola B 2-4AC > 0 Hyperbola

28 HW 8.6 Page 531, odd