The Circle

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.

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} = r^{2}

Example:
Find the equation of the circle with center (2,1) and
radius 4.
Solution:
We have:
(x  2)^{2} + (y  1)^{2} = 4^{2}
= 16
Exercise:

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

(x  2)^{2} + (y + 1)^{2} = 9

x^{2}  2x + y^{2} + 6y = 14

x^{2} + y^{2} + 4x  4y = 9

x^{2} + y^{2} + 6x + 2y = 29

Find the area between the circles:
x^{2} + y^{2}  6x + 4y = 12
and
x^{2} + y^{2}  6x + 4y = 23
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
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