In the xy-plane, a circle is represented by the equation x^2 + y^2 - 10x + 8y - 8 = 0. Wha… — SAT

In the $xy$ -plane, a circle is represented by the equation $x^2 + y^2 - 10x + 8y - 8 = 0$ . What is the diameter of this circle?

To find the diameter of the circle, we first rewrite the equation in standard form,

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

, by completing the square. Grouping the terms gives

(x^2 - 10x) + (y^2 + 8y) = 8

. Adding

25

and

16

to both sides yields

(x - 5)^2 + (y + 4)^2 = 49

. Since

r^2 = 49

, the radius of the circle is

7

. The diameter is twice the radius, which is

14

Group the

x

-terms and

y

-terms and move the constant to the right side of the equation.

(x^2 - 10x) + (y^2 + 8y) = 8

This prepares the equation for completing the square for both variables.

Complete the square for both the

x

and

y

expressions by adding

( \frac{-10}{2} )^2 = 25

and

( \frac{8}{2} )^2 = 16

to both sides of the equation.

(x^2 - 10x + 25) + (y^2 + 8y + 16) = 8 + 25 + 16

Completing the square converts the equation into the standard form of a circle's equation.

Rewrite the left side as squared binomials and simplify the right side.

(x - 5)^2 + (y + 4)^2 = 49

This puts the equation in the standard form

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

Identify

r^2

from the standard form equation and calculate the radius

r

r^2 = 49 \implies r = 7

In the standard form equation of a circle, the constant on the right side is the square of the radius.

Calculate the diameter by doubling the radius.

Diameter

= 2r = 2(7) = 14

The diameter of a circle is twice its radius.

Equations of Circles in the Coordinate Plane

Question