PreCalculus
Section 7-1 Parabolas
Name: ______________________
Period _____
Conic Sections (conics) are shown on page 422. The common conics are the parabola, ellipse, circle and
hyperbola. The general form of conic sections is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
Locus: set of all points that fulfill a geometric property
*A Parabola represents the locus of points (x, y) that are equidistant from a point (focus) and
a specific line (directrix). A parabola is symmetric about the line perpendicular to the directrix
through the focus called the axis of symmetry. The vertex is the intersection of the parabola and the
axis of symmetry.
* The standard form of the equation of a parabola with vertex at (0, 0) is as follows:
Equation
Focus
Directrix
Axis of Symmetry
x2 = 4py
(0, p)
y = _____
Vertical (_____)
y2 = 4px
(p, 0)
x = _____
Horizontal (_____)
*Remember: Focus lies inside the parabola and directrix is outside.
The distance from the focus to the vertex must equal the distance from the directrix to the vertex.
Ex. 1: Identify the vertex, focus, directrix, and axis of symmetry. Then graph the parabola.
a. x 
1 2
y
2
PreCalculus Ch 7 Notes_Page 1
1
b. y   x 2
4
You Try: Identify the vertex, focus, directrix, and axis of symmetry. Then graph the parabola.
a. x2  8 y
b. y 2  12 x
You Try: The cables of a suspension bridge are in the shape of a parabola. The towers supporting the
cable are 100 feet apart and 40 feet high. If the cables touch the road surface midway between the towers,
what is the height of the cable at a point 20 feet from the center of the bridge? (Consider the vertex is at
the origin.)
Look at the diagram on top of page 423. You can also look at where the equation for a sideways parabola
comes from. See the key concept on page 423
( x  h)2  4 p( y  k )
( y  k )2  4 p( x  h)
Ex. 2: For  y  3  8  x  1 , identify the vertex, focus, axis of symmetry, and directrix. Then graph
the parabola.
2
PreCalculus Ch 7 Notes_Page 2
You Try: For 8  y  3   x  4  , identify the vertex, focus, axis of symmetry, and directrix.
Then graph the parabola.
2
Ex. 3: The parabolic mirror for the California Institute of Technology’s Hale telescope at Mount Palomar
has a shape modeled by y 2  2668x , where x and y are measured in inches. What is the focal length of the
mirror?
To determine the characteristics of a graph, you may need to rewrite the equation in standard form. You
can sometimes do this using completing the square.
Ex. 4: Write x 2  8x  y  18 in standard form. Identify the vertex, focus, axis of symmetry, and
directrix. Then graph the parabola.
You Try: Rewrite the equation 2 y  x 2  2 x  7 in standard form. Identify the vertex, focus and directrix.
Then graph.
PreCalculus Ch 7 Notes_Page 3
Ex. 5: Write an equation for a parabola with the given characteristics. Then graph.
a.
focus (2, 1) and vertex (-5, 1)
b.
vertex (3, -2), directrix is y = -1
c. focus (-1, 7), opens up, contains (3, 7)
d. focus (1, 3), opens left, contains (-14, 11)
PreCalculus Ch 7 Notes_Page 4
PreCalculus
Section 7-2 Ellipses and Circles
Name:___________________
An ellipse is the locus of points in a plane such that the sum of the distances from two fixed points, called
foci, is constant. See the diagrams on page 432-433. The segment that contains the foci of an ellipse is
called the major axis, and the midpoint of the major axis is called the center. The segment through the
center with endpoints on the ellipse and perpendicular to the major axis is the minor axis. The two
endpoints of the major axis are the vertices, and the endpoints of the minor axis are the co-vertices.
* The standard equation of an ellipse with center at (0, 0)
Equation
Major Axis
Vertices
Co-Vertices
x2
a2
x2
b2


y2
b2
y2
a2
 1.
Horizontal
( a, 0)
(0,  b)
 1.
Vertical
(0,  a)
( b, 0)
* The major and minor axes are of lengths 2a and 2b, respectively, where a > b > 0.
The foci of the ellipse lie on the major axis at a distance of c units from the center,
where c2 = ________ (Remember, c is the distance between the center and the focus.)
Ex. 1: Identify the vertices, co-vertices, foci, and the length of the major axis of the ellipse.
Then graph and state domain and range.
a.
9x2 + 36y2 = 324.
PreCalculus Ch 7 Notes_Page 5
b.
x2 
y2
25
1
Ex 2: Write an equation of the ellipse with the given characteristics and center at (0, 0).
a. Focus: (3, 0)
b. Vertex: (0, 7)
Co-vertex: (0, 4)
Focus: (0, 3)
Key concept box on page 433: Standard Forms of Equations for Ellipses
 x  h
a
2
2
y k

b
2
2
1
 x  h
b
2
2
y k

a
2
2
1
where c2  a 2 b2
Ex. 3: Graph the ellipse. Then identify the center, foci, vertices, co-vertices, major axis and minor axis.
 x  2
9
2
 y  1

4
2
1
PreCalculus Ch 7 Notes_Page 6
Ex. 4: Write the equation in standard form for the ellipse x2  9 y 2  4 x  18 y  4  0
Then find the coordinates of the center, foci, vertices, and co-vertices.
You Try: Graph the ellipse. Then identify the center, foci, vertices, co-vertices, major axis and minor axis.
4 x2  24 x  y 2  10 y  3  0
Ex. 5: Write an equation for an ellipse with each set of characteristics.
a. major axis from (5, -2) to (-1,-2); minor axis from (2,0) to (2, -4)
PreCalculus Ch 7 Notes_Page 7
b. Write an equation for an ellipse: Vertices at (3, -4) and (3, 6); foci at (3, 4) and (3, -2)
*A circle is the set of all the points (x, y) that are equidistant from a fixed point (center).
* The standard form of the equation of a circle with center at (0, 0) and radius r is as follows:
x2  y 2  r 2
Ex. 6: Graph y2 = x2 + 16.
Identify the radius of the circle.
You Try: Graph the equation. Identify the radius
x2 = 4  y2
Ex. 7: The point (3, 4) lies on a circle whose center is the origin. Write the standard form of the
equation of the circle.
You Try: Write the standard form of the equation of the circle with center at the origin that passes
through the point (6, 3).
PreCalculus Ch 7 Notes_Page 8
Ex. 8: Write an equation of the line tangent to the circle x2 + y2 = 17 at (4, 1).
You Try: Write an equation of the line tangent to the circle x2 + y2 = 34 at (3, 5).
Standard Form of Equations of Circles with center (h, k) and radius r is  x  h    y  k   r 2
2
2
Ex. 9: Write the equation in standard form for the circle x2  y 2  6 x  10 y  2  0
Identify the center and length of the radius.
c
.
a
It will determine how “circular” or “stretched” the ellipse will be.
The formula for eccentricity of an ellipse is e 
To find c, use c 2  a 2  b 2 . Since a > c for ellipses, the value e will be between 0 and 1.
The value c represents the distance between one of the foci and the center of the ellipse.
Ex. 10: Determine the eccentricity of the ellipse given by each of the following. Round to the hundredth.
a.
 x  4
64
2
 y  3

36
2
1
PreCalculus Ch 7 Notes_Page 9
x 2  y  8
b.

1
18
48
2
Ex. 11: The eccentricity of the orbit of Uranus is 0.47. Its orbit around the sun has a major axis length of
38.36 AU (astronomical units). What is the length of the minor axis of the orbit?
Ex. 12: Write each equation in standard form. Identify the related conic.
a. x2  y 2  2 x  6 y  6  0
b. 9 x2  4 y 2  8 y  32  0
c. x2  4 x  4 y  16  0
You Try: An elliptically shaped garden is surrounded by a wood walkway. The garden is 10 meters long
and 8 meters wide. The walkway is 1 meter wide. Find the equation describing the ellipse that includes
both the garden and the walkway. (Consider the center is at the origin.)
PreCalculus Ch 7 Notes_Page 10
PreCalculus
7-3 Hyperbolas
Name:___________________
* Hyperbola: The set of all points P such that the difference of the distances between P and two
fixed points (foci), is a constant.
* A hyperbola has two axes of symmetry, which both go through the center. The transverse axis has
length of 2a, connects the vertices and intersects the hyperbola. The conjugate axis has length of 2b
and does not intersect the hyperbola. See key concept on page 443.
* The standard equation of a hyperbola with center at (0, 0)
Equation
Transverse
Asymptotes
Vertices
Axis
Horizontal
b
( a, 0)
x2 y 2
y x
.


1
a
a 2 b2
y2
a2

x2
b2
1.
Vertical
y
a
x
b
(0,  a)
* The foci lie on the transverse axis, c units from the center, where c2  a2 + b2.
Ex. 1: Graph the hyperbola and identify the center, vertices, foci, transverse axis and asymptotes.
x2 y 2
a.
b. 36y2  9x2  324

1
49 81
PreCalculus Ch 7 Notes_Page 11
Ex. 2: Write an equation of the hyperbola with foci at (5, 0), (5, 0) and vertices at (4, 0), (4, 0).
You Try: Write an equation of the hyperbola with foci at (0, 8), (0, 8) and vertices at (0, 5), (0, 5).
Ex. 3: Graph the hyperbola and identify the center, vertices, foci, transverse axis and asymptotes.
 y  4
4
2
 x  2

9
2
1
Ex. 4: Write in standard form and then graph the hyperbola.
a. 4 x2  y 2  24 x  4 y  28 .
PreCalculus Ch 7 Notes_Page 12
b. Graph. 2 x 2  3 y 2  12 x  36
Ex. 5: Write the equation for the hyperbola with the given characteristics.
a. foci (1, -5), (1,1); transverse axis length 4 units
b. vertices (-3, 10), (-3, -2); conjugate axis length 6 units
c. foci (2, -2), (12, -2); asymptotes y 
PreCalculus Ch 7 Notes_Page 13
3
29
3
13
x , y  x
4
4
4
4
c
.
a
The eccentricity of a hyperbola is always greater than 1. To find c, use a2 + b2 = c2. (so, c > a )
The formula for eccentricity for all conics is e 
Recall: The eccentricity of an ellipse is 0 < e < 1, since a2  b2 = c2. (so, a > c)
Ex. 6: Determine the eccentricity of the hyperbola given by the given equation.
a.
 y  2
32
2
 x  1

25
2
1
b.
 x  8
64
2
 y  4

80
2
1
Classify Conics using the Discriminant, B 2 4 AC : See key concept on page 447.
The graph of a second degree equation of the form Ax2  Bxy  Cy 2  Dx  Ey  F  0 is
 a circle if B 2 4 AC  0 ; B  0 and A  C
 an ellipse if B 2 4 AC  0 ; either B  0 and A  C
 a parabola if B 2 4 AC  0; either A  0 or C  0 ;
 a hyperbola if B 2 4 AC  0; A and C have the opposite sign.
Ex. 7: Use the discriminant to identify each conic section.
a. 2x2 + y2 – 2x = 12
b. 4x2 + 4y2 – 4x + 8 = 0 c. 2x + 2y2 – 6y – 10 = 0
d. 2x2 – 2x + 12 = y2
Ex. 8: LORAN (Long Range Navigation) is a navigation system for ships relying on radio pulses that is
not dependent on visibility conditions. Suppose LORAN stations E and F are located 350 miles apart
along a straight shore with E due west of F. When a ship approaches the shore, it receives radio pulses
from the stations and is able to determine that it is 80 miles farther from station F than it is from station E.
a. Find the equation for the hyperbola on which the ship is located.
b. Find the exact coordinates of the ship if it is 125 miles from the shore.
PreCalculus Ch 7 Notes_Page 14