1. Trigonometry
Chapter 9
Section 1

2. ( ) , Distance Formula d = (x2 – x1)2 + (y2 – y1)2 Midpoint Formula
( )
x1 + x2 y1 + y2 2 ,
m =

3. GUIDED PRACTICE
for Examples 1 and 2
1. What is the distance between (3, –3) and (–1, 5)?
d = (x2 – x1)2 + (y2 – y1)2

4. GUIDED PRACTICE for Examples 1 and 2
2. The vertices of a triangle are R(– 1, 3), S(5, 2), and T(3, 6). Classify ∆RST as scalene, isosceles, or equilateral.
d = (x2 – x1)2 + (y2 – y1)2
R – 1,3
S 5, 2
T 3, 6

5. ( ) , GUIDED PRACTICE for Examples 3, 4 and 5
For the line segment joining the two given points, (a) find the midpoint and (b) write an equation for the perpendicular bisector.
( )
x1 + x2 y1 + y2 2 ,
m =
3. (0, 0), (24, 12)
4. (–2, 1), (4, –7)

6. ( ) , GUIDED PRACTICE for Examples 3, 4 and 5
For the line segment joining the two given points, (a) find the midpoint and (b) write an equation for the perpendicular bisector.
( )
x1 + x2 y1 + y2 2 ,
m =
5. (3, 8), (–5, –10)

7. Solve a multi-step problem
EXAMPLE 5
Solve a multi-step problem
Asteroid Crater
Many scientists believe that an asteroid slammed into Earth about 65 million years ago on what is now Mexico’s Yucatan peninsula, creating an enormous crater that is now deeply buried by sediment. Use the labeled points on the outline of the circular crater to estimate its diameter. (Each unit in the coordinate plane represents 1 mile.)
Find the equation of the perpendicular bisectors of lines OB and OA.
Find the center point of the crater.
Find the diameter of the crater.

8. EXAMPLE 5
Solve a multi-step problem
Write equations for the perpendicular bisectors of AO and OB.

9. EXAMPLE 5
Solve a multi-step problem
Find the coordinates of the center of the circle, where AO and OB intersect, by solving the system formed by the two equations in Step 1.

10. EXAMPLE 5
Calculate the radius of the circle using the distance formula. The radius is the distance between C and any of the three given points.
d = (x2 – x1)2 + (y2 – y1)2

11. Trigonometry
Chapter 9
Section 2

12. PARABOLAS
We know that the graph of y = ax2 is a parabola.
The standard form of the equation of a parabola with vertex at (0, 0) is:
Equation Focus Directrix Axis of Symmetry
x2 = 4py (0, p) y = -p Vertical (x = 0)
y2 = 4px (p, 0) x = -p Horizontal (y = 0)

13. GUIDED PRACTICE
Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola.
1. y2 = –6x
(Make it look like the standard form)
SOLUTION
(– , 0), x = , y = 0 3 2

14. GUIDED PRACTICE
Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola.
2. x2 = 2y
(0, ), x = 0 , y = – 1 2
SOLUTION

15. GUIDED PRACTICE
Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. 14
3. y = – x2
SOLUTION
(0, –1 ), x = 0 , y = 1

16. GUIDED PRACTICE
Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola.
4. x = – y2 13
SOLUTION
(- , 0), x = , y = 0 3 4 3 4

17. GUIDED PRACTICE
Write the standard form of the equation of the parabola with vertex at (0, 0) and the given directrix or focus.
5. Directrix: y = 2
SOLUTION
x2 = – 8y
6. Directrix: x = 4
SOLUTION
y2 = – 16x

18. GUIDED PRACTICE
Write the standard form of the equation of the parabola with vertex at (0, 0) and the given directrix or focus.
7. Focus: (–2, 0)
y2 = – 8x
SOLUTION
8. Focus: (0, 3)
x2 = 12y
SOLUTION

19. Trigonometry
Chapter 9
Section 3

20. CIRCLES
The standard form of the equation of a circle with center at (0, 0) and radius r is:
x2 + y2 = r2

21. Graph the equation. Identify the radius of the circle.
GUIDED PRACTICE
for Examples 1, 2, and 3
Graph the equation. Identify the radius of the circle. 1.
x2 + y2 = 9
x2 + y2 = r2

22. GUIDED PRACTICE
for Examples 1, 2, and 3 2.
y2 = –x2 + 49
x2 + y2 = r2

23. GUIDED PRACTICE
for Examples 1, 2, and 3 3.
x2 – 18 = –y2
x2 + y2 = r2

24. GUIDED PRACTICE for Examples 1, 2, and 3
x2 + y2 = r2
4. Write the standard form of the equation of the circle that passes through (5, –1) and whose center is the origin.
5. Write an equation of the line tangent to the circle x2 + y2 = 37 at (6, 1).

25. EXAMPLE 4 Write a circular model Cell Phones
x2 + y2 = r2
Write a circular model
Cell Phones
A cellular phone tower services a 10 mile radius. You get a flat tire 4 miles east and 9 miles north of the tower. Are you in the tower’s range?
In the diagram above, the origin represents the tower and the positive y-axis represents north.

26. EXAMPLE 4
Write a circular model
STEP 2
Substitute the coordinates (4, 9) into the inequality from Step 1.

27. GUIDED PRACTICE
for Examples 4 and 5
6. WHAT IF? Suppose you drive west after fixing your tire. For how many more miles will you be in range of the tower?

28. Trigonometry
Chapter 9
Section 4

29. ELLIPSES
The standard form of the equation of an ellipse with center at (0, 0) and radius r is:
Equation Major Axis Vertices Co-Vertices
Horizontal (+/-a, 0) (0, +/-b)
Vertical (0, +/-a) (+/-b, 0)
The major axis is 2a and minor axes is 2b.
The foci are c units from the center, where c2 = a2 - b2. x2 a2 +
y2 b2 x2 b2 +
y2 a2

30. GUIDED PRACTICE
for Example 1
Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse. x2 16 +
y29 1.
= 1

31. GUIDED PRACTICE
for Example 1 x2 36 +
y2 49 2.
= 1

32. GUIDED PRACTICE
for Example 1 3.
25x2 + 9y2 = 225

33. GUIDED PRACTICE
for Examples 2, 3 and 4
Write an equation of the ellipse with the given characteristics and center at (0, 0).
4. Vertex: (7, 0); co-vertex: (0, 2)
5. Vertex: (0, 6); co-vertex: ( –5, 0)

34. GUIDED PRACTICE
for Examples 2, 3 and 4
6. Vertex: (0, 8); focus: ( 0, –3)
7. Vertex: (–5, 0); focus: ( 3, 0)

35. EXAMPLE 3
Solve a multi-step problem
Lightning
When lightning strikes, an elliptical region where the strike most likely hit can often be identified. Suppose it is determined that there is a 50% chance that a lightning strike hit within the elliptical region shown in the diagram.
• Write an equation of the ellipse.
• The area A of an ellipse is A = π ab. Find the area of the elliptical region.

36. Solve a multi-step problem
EXAMPLE 3
Solve a multi-step problem
SOLUTION
The area A of an ellipse is A = π ab.

37. GUIDED PRACTICE
for Examples 2, 3 and 4 8.
What If ? In Example 3, suppose that the elliptical region is 250 meters from east to west and meters from north to south. Write an equation of the elliptical boundary and find the area of the region.

38. Trigonometry
Chapter 9
Section 6

39. CONIC SECTIONS
The standard form of equations of Translated Conics
Circle (x – h)2 + (y – k)2 = r2
Horizontal Axis Vertical Axis
Parabola (y-k)2 = 4p(x-h) (x-h)2 = 4p(y-k)
Ellipse (x-h)2 + (y-k)2 = 1 (x-h)2 + (y-k)2 = a b b a2
Hyperbola: (x-h)2 - (y-k)2 = 1 (y-k)2 - (x-h)2 = a b a b2

40. CONIC SECTIONS Circle: center = (h, k) radius = r
Parabola: vertex = (h, k)
p is the distance from vertex to focus
Ellipse: center = (h, k)
a (always the bigger), is the distance from center to vertices
b (always the smaller), is the distance from center to co-vertices
c is the distance from center to foci, where c2 = a2 – b2
Hyperbola: center = (h, k)
a is always first, and:
- if a is under x2: horizontal, a is the distance from center to vertices and asymptotes are y = +/-(b/a)
- If a is under y2: vertical, a is the distance from center to vertices and asymptotes are y = +/-(a/b)
c is the distance from center to foci, where c2 = a2 + b2

41. GUIDED PRACTICE
for Examples 1 and 2 1.
Graph (x + 1)2 + (y – 3) 2 = 4.

42. GUIDED PRACTICE
for Examples 1 and 2 2.
Graph (x – 2)2 = 8 (y + 3) 2.

43. GUIDED PRACTICE
for Examples 1 and 2
(y – 4)2 9 3.
Graph
(x + 3)2 –
= 1
SOLUTION
hyperbola with vertices (–4, 4) and (–2, 4), asymtotes y = –2x – 2 and y = 2x + 10

44. GUIDED PRACTICE
for Examples 1 and 2
(x – 2)2
(y – 1)2 9 4.
Graph +
= 1 16

45. GUIDED PRACTICE
for Examples 3, 4 and 5 5.
Write the equation of parabola with vertex at (3, – 1) and focus at (3, 2).
The standard form of the equation is (x – 3)2 = 12(y + 1).
ANSWER 6.
Write the equation of the hyperbola with vertices at (–7,3) and (–1, 3) and foci at (–9, 3) and (1, 3).
ANSWER
The standard form of the equation is
(x + 4)2 9 –
(y – 3)2 16
= 1

46. GUIDED PRACTICE
for Examples 3, 4 and 5
Identify the line(s) of symmetry for the conic section.
(x – 5)2 64
(y)2 16 7. +
= 1

47. GUIDED PRACTICE
for Examples 3, 4 and 5
Identify the line(s) of symmetry for the conic section. 8.
(x + 5)2 = 8(y – 2).

48. GUIDED PRACTICE
for Examples 3, 4 and 5
Identify the line(s) of symmetry for the conic section. 9.
(x – 1)2 49
(y – 2)2 –
= 1
121
x = 1 and y = 2.
ANSWER

49. GUIDED PRACTICE
for Examples 6 and 7
10.
Classify the conic given by x2 + y2 – 2x + 4y + 1 = 0. Then graph the equation.

50. GUIDED PRACTICE
for Examples 6 and 7
11.
Classify the conic given by 2x2 + y2 – 4x – 4 = 0. Then graph the equation.

51. GUIDED PRACTICE
for Examples 6 and 7
12.
Classify the conic given by y2 – 4y2 – 2x + 6 = 0. Then graph the equation.

52. GUIDED PRACTICE
for Examples 6 and 7
13.
Classify the conic given by 4x2 – y2 – 16x – 4y – 4 = 0. Then graph the equation.
ANSWER
Hyperbola
(x –2)2 4
(y +2)2 16
= 1 –

53. GUIDED PRACTICE
for Examples 6 and 7
14.
Astronomy
An asteroid’s path is modeled by 4x y2 – 12x – 16 = 0 where x and y are in astronomical units from the sun. Classify the path and write its equation in standard form. Then graph the equation.
ANSWER
Hyperbola
4(x – 1.5)2 4 – y2
= 1

54. Trigonometry
Chapter 9
Section 7

55. GUIDED PRACTICE
for Examples 1 and 2
1. x2 + y2 = 13
y = x – 1 2.
x2 + 8y2 – 4 = 0
y = 2x + 7 3.
y2 + 6x – 1 = 0
y = –0.4x + 2.6

56. GUIDED PRACTICE
for Examples 1 and 2 4.
y = 0.5x – 3
x2 + 4y2 – 4 = 0 5.
y2 – 2x – 10 = 0
y = x 1 – 6.
y = 4x – 8
9x2 – y2 – 36 = 0

57. GUIDED PRACTICE
for Examples 3 and 4
Solve the system. 7.
–2y2 + x + 2 = 0
x2 + y2 – 1 = 0 8.
x2 + y2 – 16x + 39 = 0
x2 – y2 – 9 = 0

58. GUIDED PRACTICE
for Examples 3 and 4
Solve the system. 9.
x2 + 4y2 + 4x + 8y = 8
y2 – x + 2y = 5

59. GUIDED PRACTICE for Examples 3 and 4 10.
WHAT IF? In Example 4, suppose that a ship’s LORAN system locates the ship on the two hyperbolas whose equations are given below. Find the ship’s location if it is south of the x-axis.
x2 – y2 – 12x + 18 = 0
Equation 1
y2 – x2 – 4y + 2 = 0
Equation 2