### Chapter 9

```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
2
2
2
c =a +b
Example
8
6
D
4
2
E
-5
F
5
-2
Find the
distance
between
point D and
point F.
10
Distance Formula
D = √(x2 –
2
x1)
+ (y2 –
2
y1)
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
2
2
2
(x – h) + (y – k) = r
Example
Find
an equation of the
circle with center (-2,5)
2
2
(x + 2) + (y – 5) = 9
Translation
Sliding
a graph to a new
position in the
coordinate plane
without changing its
shape
Translation
8
6
4
2
-10
-5
5
-2
10
Example
6
4
Graph (x –
-10
2
2)
2
-5
+ (y +
5
-2
-4
-6
-8
-10
2
6)
10
=4
Example
If
the graph of the
equation is a circle, find
its center and radius.
2
2
x + y + 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 =
2
a(x-h)
Vertex (h,k) symmetry x
x-h=
=h
2
a(y-k)
Vertex (h,k) symmetry y
=k
Equation of a Parabola
Remember:
y – k = a(x –
 (h,k) is the vertex of the
parabola
2
h)
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)
8
Example 1
6
4
2
Vertex (- 5,1)
-5
dir ectrix (y = -2)
5
-2
-4
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 –
a) a
2
h)
= 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 –
c) Vertex
2
h)
(h, k)
d) Focus (h, k+c)
e) Directrix y = k – c
f) Axis of Symmetry x = h
x - h = a(y
a) a
2
–k)
= 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 –
c) Vertex
2
k)
(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:
2
y – 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 = 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
2
denominator of y 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
2
2
2
equation b =a – c
Example 2
Graph the ellipse
2
2
4x + y = 64
and find its foci
Example 3

Find an equation of an
ellipse having x-intercepts
√2 and - √2 and yintercepts 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 = ± 6
 Simplify to get x2 - y2 = 1
9 16
Graphing
The graph has two xintercepts and no yintercepts
 (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, 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
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.
2
2
x – 4y – 2x – 16y – 11 = 0
```