*Section 6-2: Equations
of Circles*
* *
__Definition of a Circle__
* *
*A circle
is the set of all points in a plane equidistant from a fixed point called
the center point.
We can derive the equation directly from the distance formula. If
we place the center point on the origin point, the equation of a circle
with center point (0, 0) and radius r is:*
*x*^{2} + y^{2}
= r^{2}
*Look at the family of
circles drawn:*
*A simple translation
of the circle equation becomes:*
*(x - h)*^{2} +
(y - k)^{2} = r^{2}
*With center at (h, k)
and radius r.*
*Here are some examples:*

__Sample Problems__
* *
*1) Find the center, radius
and graph the equation: (x - 2)*^{2}
+ (y + 5)^{2} = 17

* *

*
Solution: Center point : (2, -5),
radius =
The graph is below:*
*2) x*^{2}
+ y^{2} - 8x + 4y - 8
= 0 Find center, radius and graph.

* *

*
Solution: We need to put the equation
into the correct form. We will do this by completing the square!!*

*
(x*^{2} - 8x
) + (y^{2} + 4y
) = 8 Complete the square! See 1.6

*
(x*^{2} - 8x +
16) + (y^{2}
+ 4y + 4)
= 8 + 16 + 4

*
(x - 4)*^{2}
+ (y + 2)^{2} = 28
Now it's in the correct form!!

*
Center point (4, -2) with radius =
The graph is:*

* *
*3) Find the intersection
of the line y = x - 1 and the circle x*^{2}
+ y^{2} = 25.
* *

*
Solution: A line could intersect a circle
twice or once (if it is tangent) or not intersect at all. To find
the intersection use substitution. Replace y in the circle equation
with x - 1.*

* *

*
x*^{2} + (x - 1)^{2}
= 25

*
x*^{2} + x^{2}
- 2x + 1 = 25

*
2x*^{2} - 2x - 24 = 0

*
x*^{2} - x - 12 = 0
Now factor

*
(x - 4)(x + 3) = 0*

*
x = 4 or x = -3*

*
Substituting back in to find y gives the following points:*

*
(4, 3) and (-3, -4) If you check these points in both equations,
you will discover they make both true. The graph of the system is:*
*What would it mean if
the solutions were imaginary?*

*4) Sketch the graph
of *
* *
*
Solution: The above graph is part of
a circle. Why? Because it only includes the positive values
for y. Thus it is the top half of a circle with radius 4. It
is a semi-circle. By only studying this part of the circle, it makes
it a function. It now passes the vertical line test. Circles
are not functions! Here is the graph:*

*That's it for circles.
Let's head toward Ellipses!*