Equation of a Circle From Center and Radius

Equation of a Circle from Center and Radius

A circle is the set of all points at a fixed distance from its center. If the center is ((h,k)) and the radius is (r), the standard equation is

[ (x-h)^2+(y-k)^2=r^2. ]

Method

1. Identify the center ((h,k)).
   Be careful with signs: a center of ((3,-2)) gives ((x-3)^2) and ((y+2)^2).

2. Identify the radius (r).
   Use the given radius directly, then square it to get (r^2).

3. Substitute into the formula.
   Write the equation with the correct center values and the squared radius.

4. Simplify the final answer.
   If needed, reduce fractions or compute the square of the radius.

Check

• The equation should have both squared terms ((x-h)^2) and ((y-k)^2).
• The right-hand side must be a nonnegative number, since it is (r^2).
• If you expand mentally, the center should match the shifts in the squared terms.

Common mistake to avoid

Do not change the signs inside the parentheses. The sign in the equation is always the opposite of the sign in the center coordinates.