Chapter 9

- Analytic Geometry

Section 9-1

- Distance and Midpoint Formulas

Pythagorean Theorem

- If the length of the hypotenuse of a right

triangle is c, and the lengths of the other two

sides are a and b, then c2 a2 b2

Example

Find the distance between point D and point F.

Distance Formula

- D v(x2 x1)2 (y2 y1)2

Example

- Find the distance between points A(4, -2) and

B(7, 2) - d 5

Midpoint Formula

- M( x1 x2, y1 y2)
- 2 2

Example

- Find the midpoint of the segment joining the

points (4, -6) and (-3, 2) - M(1/2, -2)

Section 9-2

- Circles

Conics

- Are obtained by slicing a double cone
- Circles, Ellipses, Parabolas, and Hyperbolas

Equation of a Circle

- The circle with center (h,k) and radius r has the

equation - (x h)2 (y k)2 r2

Example

- Find an equation of the circle with center (-2,5)

and radius 3. - (x 2)2 (y 5)2 9

Translation

- Sliding a graph to a new position in the

coordinate plane without changing its shape

Translation

Example

- Graph (x 2)2 (y 6)2 4

Example

- If the graph of the equation is a circle, find

its center and radius. - x2 y2 10x 4y 21 0

Section 9-3

- Parabolas

Parabola

- A set of all points equidistant from a fixed line

called the directrix, and a fixed point not on

the line, called the focus

Vertex

- The midpoint between the focus and the directrix.

Parabola - Equations

- y-k a(x-h)2
- Vertex (h,k) symmetry x h
- x - h a(y-k)2
- Vertex (h,k) symmetry y k

Equation of a Parabola

- Remember
- y k a(x h)2
- (h,k) is the vertex of the parabola

Example 1

- The vertex of a parabola is (-5, 1) and the

directrix is the line y -2. Find the focus of

the parabola. - (-5 4)

Example 1

Example 2

- Find an equation of the parabola having the point

F(0, -2) as the focus and the line x 3 as the

directrix.

y k a(x h)2

- a 1/4c where c is the distance between the

vertex and focus - Parabola opens upward if agt0, and downward if alt

0

y k a(x h)2

- Vertex (h, k)
- Focus (h, kc)
- Directrix y k c
- Axis of Symmetry x h

x - h a(y k)2

- a 1/4c where c is the distance between the

vertex and focus - Parabola opens to the right if agt0, and to the

left if alt 0

x h a(y k)2

- Vertex (h, k)
- Focus (h c, k)
- Directrix x h - c
- Axis of Symmetry y k

Example 3

- Find the vertex, focus, directrix , and axis of

symmetry of the parabola - y2 12x -2y 25 0

Example 4

- Find an equation of the parabola that has vertex

(4,2) and directrix y 5

Section 9-4

- Ellipses

Ellipse

- The set of all points P in the plane such that

the sum of the distances from P to two fixed

points is a given constant.

Focus (foci)

- Each fixed point
- Labeled as F1 and F2
- PF1 and PF2 are the focal radii of P

Ellipse- major x-axis

Ellipse- major y-axis

Example 1

- Find the equation of an ellipse having foci

(-4, 0) and (4, 0) and sum of focal radii 10.

Use the distance formula.

Example 1 - continued

- Set up the equation
- PF1 PF2 10
- v(x 4)2 y2 v(x 4)2 y2 10
- Simplify to get x2 y2 1
- 25 9

Graphing

- The graph has 4 intercepts
- (5, 0), (-5, 0), (0, 3) and (0, -3)

Symmetry

- The ellipse is symmetric about the x-axis if the

denominator of x2 is larger and is symmetric

about the y-axis if the denominator of y2 is

larger

Center

- The midpoint of the line segment joining its foci

General Form

- x2 y2 1
- a2 b2
- The center is (0,0) and the foci are (-c, 0) and

(c, 0) where - b2 a2 c2
- focal radii 2a

General Form

- x2 y2 1
- b2 a2
- The center is (0,0) and the foci are (0, -c) and

(0, c) where - b2 a2 c2
- focal radii 2a

Finding the Foci

- If you have the equation, you can find the foci

by solving the equation b2 a2 c2

Example 2

- Graph the ellipse
- 4x2 y2 64
- and find its foci

Example 3

- Find an equation of an ellipse having

x-intercepts v2 and - v2 and y-intercepts 3 and

-3.

Example 4

- Find an equation of an ellipse having foci (-3,0)

and (3,0) and sum of focal radii equal to 12.

Section 9-5

- Hyperbolas

Hyperbola

- The set of all points P in the plane such that

the difference between the distances from P to

two fixed points is a given constant.

Focal (foci)

- Each fixed point
- Labeled as F1 and F2
- PF1 and PF2 are the focal radii of P

Example 1

- Find the equation of the hyperbola having foci

(-5, 0) and (5, 0) and difference of focal

radii 6. Use the distance formula.

Example 1 - continued

- Set up the equation
- PF1 - PF2 6
- v(x 5)2 y2 - v(x 5)2 y2 6
- Simplify to get x2 - y2 1
- 9 16

Graphing

- The graph has two x-intercepts and no

y-intercepts - (3, 0), (-3, 0)

Asymptote(s)

- Line(s) or curve(s) that approach a given curve

arbitrarily, closely - Useful guides in drawing hyperbolas

Center

- Midpoint of the line segment joining its foci

General Form

- x2 - y2 1
- a2 b2
- The center is (0,0) and the foci are (-c, 0) and

(c, 0), and difference of focal radii 2a where b2

c2 a2

Asymptote Equations

- y b/a(x) and
- y - b/a(x)

General Form

- y2 - x2 1
- a2 b2
- The center is (0,0) and the foci are (0, -c) and

(0, c), and difference of focal radii 2a where b2

c2 a2

Asymptote Equations

- y a/b(x)
- and
- y - a/b(x)

Example 2

- Find the equation of the hyperbola having foci

(3, 0) and (-3, 0) and difference of focal

radii 4. Use the distance formula.

Example 3

- Find an equation of the hyperbola with

asymptotes - y 3/4x and y -3/4x and foci (5,0) and

(-5,0)

Section 9-6

- More on Central Conics

Ellipses with Center (h,k)

- Horizontal major axis (x h)2

(y-k)2 1 - a2 b2
- Foci at (h-c,k) and (h c,k) where c2 a2 - b2

Ellipses with Center (h,k)

- Vertical major axis
- (x h)2 (y-k)2 1
- b2 a2
- Foci at (h, k-c) and (h,c k) where c2 a2 - b2

Hyperbolas with Center (h,k)

- Horizontal major axis (x h)2 -

(y-k)2 1 - a2 b2
- Foci at (h-c,k) and (h c,k) where c2 a2 b2

Hyperbolas with Center (h,k)

- Vertical major axis
- (y k)2 - (x-h)2 1
- a2 b2
- Foci at (h, k-c) and (h, kc) where c2 a2 b2

Example 1

- Find an equation of the ellipse having foci

(-3,4) and (9, 4) and sum of focal radii 14.

Example 2

- Find an equation of the hyperbola having foci
- (-3,-2) and (-3, 8) and difference of focal radii

8.

Example 3

- Identify the conic and find its center and foci,

graph. - x2 4y2 2x 16y 11 0