8.7 solve quadratic systems p. 534 how do you find the points of intersection of conics?

8.7 Solve Quadratic Systems

p. 534

How do you find the points of intersection of conics?

How Many Points of Intersection?

Circle & line

Circle and parabola

Circle & ellipse

Circle & hyperbola

How Many Points of Intersection?

How Many Points of Intersection?

Ellipse & hyperbola

How Many Points of Intersection?

Hyperbola & line

Find the points of intersection of the graphs of x2 + y2 = 13 and y = x + 1.

Left side: substitute x = 2 right side: x = −3 into one of the equations and solve for y.

The points of intersection are (2,3) and (−2, −3).

x2 + y2 = 13

x2 + (x + 1)2 = 13

x2 + x2 + 2x + 1 = 13

2x2 + 2x − 12 = 0

2(x − 2)(x + 3) = 0

x = 2 or x = −3

Solve the system using substitution.

x2 + y2 = 10 Equation 1

y = – 3x + 10 Equation 2

SOLUTION

Substitute –3x + 10 for y in Equation 1 and solve for x.x2 + y2 = 10

x2 + (– 3x + 10)2 = 10x2 + 9x2 – 60x + 100 = 10

10x2 – 60x + 90 = 0x2 – 6x + 9 = 0

(x – 3)2 = 0x = 3

Equation 1

Substitute for y.Expand the power.

Combine like terms.Divide each side by 10.Perfect square trinomial

Zero product property

y = – 3(3) + 10 = 1

To find the y-coordinate of the solution, substitute x = 3 in Equation 2.

The solution is (3, 1).

ANSWER

The solution is (3, 1).

5.

y2 – 2x – 10 = 0 y = x 1 – –

SOLUTION

Substitute – x – 1 for y in Equation 1 and solve for x.y2 – 2x2 – 10 = 0

(– x – 1)2 – 2x – 10 = 0x2 + 1 + 2x – 2x – 10 = 0

x2 – 9 = 0x2 = 9

Equation 1

Substitute for y.Expand the power.

Combine like terms.Add 9 to each side.

x = ±3 Simplify.To find the y-coordinate of the solution, substitute x = −3 and x = 3 in equation 2.

y = −(–3) –1 = 2 y = −(3) –1 = −4

The solutions are (–3, 2), and (3, –4)

ANSWER

Find the points of intersection of the graphs in the system.

x2 + 4y2 − 4 = 0 (ellipse)

−2y2 + x + 2 = 0 (parabola)

Solve for x

x = 2y2 − 2

Substitute

(2y2 − 2)2 +4y2 −4 = 0

4y4 −8y2 + 4 + 4y2 − 4 = 0

4y4 −4y2 = 0

4y2(y2 −1) = 0

4y2(y +1)(y −1) = 0

4y2 = 0, y +1 = 0, y −1 = 0

y = 0, y = −1, y = 1

Left side: find x for y = −1

Right side: find x for y = 1

Solution:

(−2, 0), (0, 1), (0, −1)

SOLUTION

4.

y = 0.5x – 3

x2 + 4y2 – 4 = 0

Substitute 0.5x – 3 for y in Equation 2 and solve for x.x2 + 4y2 – 4 = 0x2 + 4 (0.5x – 3)2 – 4 = 0x2 + y (0.25x2 – 3x + 9) – 4 = 02x2 – 12x + 32 = 0x2 – 6x + 16 = 0

Equation 2

Substitute for y.Expand the power.

Combine like terms.Divide each side by 2.

This equation has no solution.

Find the points of intersection of the graphs in the system.

x2 + y2 −16x + 39 = 0

x2 − y2 −9 = 0

Eliminate y2 by adding

x2 + y2 −16x + 39 = 0

x2 − y2 −9 = 0

2x2 −16x + 30 = 0

2(x2 −8x + 15) = 0

2(x −5)(x −3) = 0

x = 3 or x = 5

Left: find y for x = 3

Right: find y for x = 5

Graphs intersect at:

(3, 0), (5, 4), (5,−4)

Solve the system by elimination.9x2 + y2 – 90x + 216 = 0 Equation 1 x2 – y2 – 16 = 0 Equation 2

SOLUTION

9x2 + y2 – 90x + 216 = 0 x2 – y2 – 16 = 0

10x2 – 90x + 200 = 0 Add.x2 – 9x + 20 = 0 Divide each side by 10.

(x – 4)(x – 5) = 0 Factorx = 4 or x = 5 Zero product property

Add the equations to eliminate the y2 - term and obtain a quadratic equation in x.

When x = 4, y = 0. When x = 5, y = ±3.ANSWER

The solutions are (4, 0), (5, 3), and (5, 23), as shown.

Navigation

A ship uses LORAN (long-distance radio navigation) to find its position.Radio signals from stations A and B locate the ship on the blue hyperbola, and signals from stations B and C locate the ship on the red hyperbola. The equations of the hyperbolas are given below. Find the ship’s position if it is east of the y - axis.

x2 – y2 – 16x + 32 = 0 Equation 1– x2 + y2 – 8y + 8 = 0 Equation 2

x2 – y2 – 16x + 32 = 0 Equation 1– x2 + y2 – 8y + 8 = 0 Equation 2

SOLUTION

STEP 1 Add the equations to eliminate the x2 - and y2 - terms.

x2 – y2 – 16x + 32 = 0– x2 + y2 – 8y + 8 = 0

– 16x – 8y + 40 = 0 Add.

y = – 2x + 5 Solve for y.

STEP 2 Substitute – 2x + 5 for y in Equation 1 and solve for x.

x2 – y2 – 16x + 32 = 0 Equation 1x2 – (2x + 5)2 – 16x + 32 = 0

3x2 – 4x – 7 = 0Substitute for y.Simplify.

(x + 1)(3x – 7) = 0 Factor.

Zero product propertyx = – 1 or x =73

ANSWER

Because the ship is east of the y - axis, it is at

STEP 3

• How do you find the points of intersection of conics?

Use substitution or linear combination to solve for the point(s) of intersection

8-7Assignment

Page 537, 9-15 odd, 23-27

(Quadratic formula will be helpful with #11)