The Circle

1. Conic Sections

A conic section is formed by intersecting a plane with a cone.  The different possible conic sections are the circle, parabola, ellipse, and the hyperbola.

2. Circles

A circle is the set of points in a plane a fixed distance from a point.  By the Pythagorean Theorem, we have that the distance r from the center (h,k) of the circle to a point (x,y) on the circle is

r = [(x - h)2 + (y - k)2]1/2

or

 (x - h)2 + (y - k)2 = r2

Example:

Find the equation of the circle with center (2,1) and  radius 4.

Solution:

We have:

(x - 2)2 + (y - 1)2 = 42 = 16

Exercise:

1. Find the equation of the circle with center (1,3) and passing through the point (7,11)

Graph the following:

2. (x - 2)2 + (y + 1)2 = 9

3.  x2 - 2x + y2 + 6y = 14

4.  x2 + y2 + 4x - 4y = 9

5.  x2 + y2 + 6x + 2y = 29

6. Find the area between the circles:

x2 + y2 - 6x + 4y = 12

and

x2 + y2 - 6x + 4y = 23

3. Example:  Circles and Tangent Lines

Find the equation of the circle that has center (3,-2) and is tangent to the line

x + 2y = 4

Solution

Since the line segment joining the center of the circle and the point where the line meets the circle is perpendicular to the line

x + 2y = 4

this segment has slope equal to the negative reciprocal of the slope of

x + 2y = 4

or

y = -1/2 x + 2

Hence this segment has slope equal to 2.  The segment lies on the line

y + 2 = 2(x - 3)

or

y = 2x - 8

The point of tangency is given by the intersection of the tangent line with this segment:

-1/2 x + 2 = 2x - 8

so

10 = 2x + 1/2 x

or

20 = 4x + x = 5x

hence

x = 4     and     y = 2(4)- 8 = 0.
Now use the distance formula to find the radius of the circle:

r = [(0 - -2)2 + (4 - 3)2]1/2 =

The equation of the circle is

(x - 3)2 + (y + 2)2 = 5