Math 35 (Spring ’09)
10.1 "The Circle and the Parabola"
Objectives:
*
Graph equations of circles written in standard form.
*
Write the equation of a circle, by given its center and radius.
*
Convert the general form of the equation of a circle to standard form.
*
Convert the general form of the equation of a parabola to standard form to graph it.
Preliminaries:
The curves formed by the intersection of a plane with an in…nite right-circular cone are called conic sections. Those
curves have four basic shapes, called circles, parabolas, ellipses, and hyperbolas. Conic sections have many applications.
For example:
Parabolas can be rotated to generate dish-shaped surfaces called paraboloids. Any light or sound placed at the focus
of a paraboloid is re‡ected outward in parallel paths. This property makes parabolic surfaces ideal for ‡ashlight and headlight
re‡ectors. It also makes parabolic surfaces good antennas. Parabolic mirrors are capable of concentrating the rays of the sun
at a single point. In architecture, many arches are parabolic in shape, because this gives them strength. Cables that support
suspension bridges hang in the shape of a parabola.
Ellipses have optical and acoustical properties that are useful in architecture and engineering. The planets and many
comets have elliptical orbits. Certain gears have elliptical shapes to provide nonuniform motion.
Hyperbolas serve as the basis of a navigational system known as LORAN (LOng RAnge Navigation). They are also
used to …nd the source of a distress signal, are the basis for the design of hypoid gears, and describe the orbits of some
comets.
Graph Equations of Circles Written in Standard Form
De…nition:
"Circle"
kA circle is the set of all points in a plane that are a …xed distance (the radius) from a …xed point called its center. k
Equation of a Circle:
The standard form of the equation of a circle with radius r and center at (h; k) is
Example 1: (Graphing circles)
Find the center and radius of each circle and then graph it:
2
2
a)
(x 2) + (y + 1) = 9
y
b)
2
(x + 3) + y 2 = 4
y
8
8
6
6
4
4
2
2
-8 -6 -4 -2
-2
2
4
6
8
-8 -6 -4 -2
-2
x
-4
-4
-6
-6
-8
-8
Page: 1
2
4
6
8
x
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
10.1
Write the Equation of a Circle, Given its Center and Radius
Example 2: (Writing the equation of a circle)
Write the equation of the circle with the following radius and center:
a)
r = 1 and center at (6; 5)
b)
r = 5 and center at ( 7; 1)
Convert the General Form of the Equation of a Circle to Standard Form
Equation of a Circle:
The general form of the equation of a circle is
Example 3: (Convert the general form of the equation of a circle to standard form)
Write the following equations of the circle in standard form
a)
x2 + y 2 4x + 2y 11 = 0
b)
x2 + y 2 + 8x + 2y =
13
Convert the General Form of the Equation of a Parabola to Standard Form to Graph it
De…nition:
"Parabola"
A parabola is the set of all points in a plane that are equidistant from a …xed point,
called the focus, and a …xed line, called the directrix.
Equation of a Parabola
The general forms of the equation of a parabola are:
1:
The graph opens upward if a > 0 and downward if a < 0
The graph opens to the right if a > 0 and to the left if a < 0
2:
Page: 2
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
10.1
Example 4: (Graphing parabolas opening right/left)
Graph
x = 12 y 2
y
6
4
2
-6
-4
-2
2
4
6
-2
x
-4
-6
Example 5: (Convert the general from of the equation of a parabola to standard form)
Write x = 3y 2 12y 13 in standard form and graph it
y
6
4
2
-6
-4
-2
2
-2
4
6
x
-4
-6
Page: 3
Notes by Bibiana Lopez