Rewrite circle equations by completing the square. | Circles | SAT Math

Use completing the square to turn expanded circle equations into center-radius form. That reveals the center and radius directly.

核心知识

When a circle equation is given in expanded form like $𝑥 2 + 𝑦 2 + 𝐷 𝑥 + 𝐸 𝑦 + 𝐹 = 0$ , complete the square on both $𝑥$ and $𝑦$ to convert it to standard form and reveal the center and radius.

深入理解

Rule: Complete the square to turn expanded form into center-radius form.

Group the $𝑥$ -terms and $𝑦$ -terms.
Complete the square on both variables.
Read the center from the signs and the radius from the right side.

If the right side is not a positive square, the circle is not real.

分步讲解

Group $𝑥$ terms and $𝑦$ terms; move the constant to the right side.
For each group, take half the linear coefficient, square it, and add to both sides.
Factor each group into a perfect square: $(𝑥 + h a l f) 2$ .
The right side is $𝑟 2$ ; take the square root to get $𝑟$ .

常见误解

Forgetting to add the completing-the-square constant to both sides of the equation.
Taking half the coefficient but forgetting to square it.
Not factoring out a leading coefficient before completing the square (e.g., $2 𝑥 2 + 2 𝑦 2 + . . .$ — divide everything by 2 first).

题目

示例解析

What are the center and radius of the circle $𝑥 2 + 𝑦 2 - 8 𝑥 + 6 𝑦 = 0$ ?

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