A circle in the xy-plane is defined by the equation (x - 7)^2 + (y + 2)^2 = 36. What is th… — SAT

A circle in the $xy$ -plane is defined by the equation $(x - 7)^2 + (y + 2)^2 = 36$ . What is the diameter of this circle?

The standard equation of a circle is

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

, where

(h, k)

is the center of the circle and

r

is the radius. In the equation

(x - 7)^2 + (y + 2)^2 = 36

, the value of

r^2

is 36. Taking the square root gives the radius,

r = 6

. The diameter of a circle is twice its radius, so the diameter is

2(6) = 12

Identify the value of the radius squared,

r^2

, from the given equation of the circle.

r^2 = 36

In the standard equation of a circle,

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

, the constant on the right side of the equation is equal to the square of the radius.

Determine the radius,

r

, by taking the square root of

r^2

r = \sqrt{36} = 6

Since the radius represents a distance, it must be the positive square root of the constant term.

Calculate the diameter of the circle by multiplying the radius by 2.

Diameter =

2 \times 6 = 12

The diameter of a circle is defined as twice its radius.

Identifying circle properties from its standard equation form

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

Soru