Circle equation from centre and radius
(x−h)²+(y−k)²=r² is the definition written in algebra.
Key formulas
Read the centre
Centre (h,k), radius r. Expanding is optional; completed-square form is often clearer.
Example
Centre (2,−3), r=5 → (x−2)²+(y+3)²=25.
General form
x²+y²+Dx+Ey+F=0 can be rewritten by completing the square to reveal centre and radius.
Worked example
Worked example
Problem. Write the equation of the circle with centre (2, −3) and radius 5.
Solution. (x − 2)² + (y + 3)² = 25.
Point test
A point on the circle satisfies equality; inside gives < r² for the left side when written as distance².
Review quiz
Centre (−1,4), radius 3. Equation is:
Practice questions
- Write the equation with centre (−1,4) and radius 3.
- Expand (x−2)²+(y+3)²=25 into general form.
Study notes
Is r² on the right always positive?
Yes — a radius of zero is a point circle; negative is impossible.