10-8 Skills Practice: Equations of Circles Explained Step by Step
The equations of circles are one of the most practical results in coordinate geometry, turning a curved shape into a single algebraic statement you can graph, test, and manipulate. In a typical 10-8 Skills Practice assignment, you are asked to write the equation of a circle from a center and radius, graph a circle from its equation, rewrite equations using completing the square, and apply circles to tangent lines and intersection problems. This guide walks through every one of those skills with worked examples, the reasoning behind each step, and the small mistakes that cost the most points.
Introduction: What an Equation of a Circle Actually Says
A circle is not defined by "roundness.But " It is defined by a fixed distance. In real terms, a circle is the set of all points that lie exactly r units away from a single point called the center. That definition is the whole source of the formula.
If the center sits at the origin, and a point (x, y) lies on the circle, the distance formula gives:
- √((x − 0)² + (y − 0)²) = r
Square both sides and you get x² + y² = r². Move the center to any point (h, k) and you get the standard form of the equation of a circle:
(x − h)² + (y − k)² = r²
Read this formula literally, and every problem becomes easier:
- (h, k) is the center.
- r
² is the radius squared. Notice the minus signs inside the parentheses: a center at (−3, 4) becomes (x + 3)² + (y − 4)² = r², not (x − 3)². That sign flip is the single most common error on this assignment.
Skill 1: Write the Equation from the Center and Radius
When you are given the center (h, k) and the radius r, substitute directly into standard form and square the radius Not complicated — just consistent. Less friction, more output..
Example: Write the equation of a circle with center (2, −5) and radius 7.
- h = 2, k = −5, r = 7
