Chapter 9

Analytic Geometry

Section 9-1Distance 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 8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

F

D

E

Find the distance between point D and point F.

Distance Formula

D = √(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

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

Example

Graph (x – 2)2 + (y + 6)2 = 4

6

4

2

-2

-4

-6

-8

-10

-10 -5 5 10

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 8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

directrix (y = -2)

f x = -2

Vertex (-5,1)

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) a = 1/4c where c is the distance between the vertex and focus

b) Parabola opens upward if a>0, and downward if a< 0

y – k = a(x – h)2

c) Vertex (h, k)

d) Focus (h, k+c)

e) Directrix y = k – c

f) Axis of Symmetry x = h

x - h = a(y –k)2

a) a = 1/4c where c is the distance between the vertex and focus

b) Parabola opens to the right if a>0, and to the left if a< 0

x – h = a(y – k)2

c) Vertex (h, k)

d) Focus (h + c, k)

e) Directrix x = h - c

f) 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

drag

Ellipse- major y-axis

drag

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

√(x + 4)2 + y2 + √(x – 4)2 + y2 = 10Simplify 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 √2 and - √2 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

√(x + 5)2 + y2 - √(x – 5)2 + y2 = ± 6Simplify 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 9-6More 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, k+c) 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