The Circle

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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 = 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:

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
x² - 2x + y² + 6y = 14
x² + y² + 4x - 4y = 9
x² + y² + 6x + 2y = 29
Find the area between the circles:

x² + y² - 6x + 4y = 12

and

x² + y² - 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 = \( \sqrt{5}\)

The equation of the circle is

        (x - 3)² + (y + 2)² = 5

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