What is the area, in square units, of the circle in the xy-plane represented by the equati… — SAT

What is the area, in square units, of the circle in the $xy$ -plane represented by the equation $x^2 + y^2 + 6x - 4y - 12 = 0$ ?

The correct answer is

25\pi

To find the area of the circle, the equation is rewritten in the standard form

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

. Rearranging the terms gives

(x^2 + 6x) + (y^2 - 4y) = 12

. Completing the square by adding 9 and 4 to both sides yields

(x + 3)^2 + (y - 2)^2 = 12 + 9 + 4 = 25

. In this standard form, the squared radius of the circle is

r^2 = 25

. Because the area of a circle is calculated as

A = \pi r^2

, the area is

25\pi

Group the

x

-terms and

y

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

(x^2 + 6x) + (y^2 - 4y) = 12

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

Complete the square for the

x

-terms by adding

(6/2)^2 = 9

to both sides, and for the

y

-terms by adding

(-4/2)^2 = 4

to both sides.

(x^2 + 6x + 9) + (y^2 - 4y + 4) = 12 + 9 + 4

, which simplifies to

(x + 3)^2 + (y - 2)^2 = 25

This expresses the equation in the standard form of a circle,

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

, where the center is

(h, k)

and the radius is

r

Identify the squared radius

r^2

from the standard form and calculate the area of the circle.

r^2 = 25

, so the area of the circle is

\pi r^2 = 25\pi

The area of a circle is calculated using the formula

A = \pi r^2

, so we directly substitute the value of

r^2

as 25.

Converting the general equation of a circle to standard form to find its radius and calculate its area.

Soru