Find the radius of each circle?

Question

Determine the center and radius of each circle.

y^2= 25 - (x+1)^2

(x - 2)^2 = 8 - (y +2)^2

Could you show me the steps on how to solve it. I don't understand thanks.

Answer 1

The center of the first circle is at (-1, 0), and its radius is 5.  The center of the second circle is at (2, -2), and its radius is √8 = 2√2.

The general form for the equation of any circle is this:

(x - h)² + (y - k)² = r², where h and k are the x and y coordinates of the center of the circle, and r is the radius of the circle.  If the h or k or both coordinates are negative numbers, then x - h = x - (-h) = x + h, or y - k = y - (-k) = y + k, or both.

To determine the center and radius of the circle, compare the given equation to the general equation given above.  For the first equation, first move the term containing the x variable to the left side of the equation::

y² = 25 - (x + 1)² -----> (x + 1)² + y² = 25

Now the equation above is the same as [x - (-1)]² + (y - 0)² = 25.  Its center is at (-1, 0), and its radius is 5 units, since 25 = (5)².

Let's look at the second equation now:

(x - 2)² = 8 - (y +2)²

This time move the term containing the y variable to the left side of the equation to get this:

(x - 2)² + (y + 2)² = 8.

The equation above is the same as (x - 2)² + [y - (-2)]² = 8.  So the center is at (2, -2), and the radius of the circle is √8 = 2√2 units.