The equation of a circle in the xy-plane is (x - 8)^2 + (y + 1)^2 = 81. What are the coord… — SAT

The equation of a circle in the $xy$ -plane is $(x - 8)^2 + (y + 1)^2 = 81$ . What are the coordinates of the center of the circle?

The center of the circle is

(8, -1)

The standard form of the equation of a circle in the

xy

-plane is

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

, where the center is

(h, k)

. In the given equation

(x - 8)^2 + (y + 1)^2 = 81

, comparing the terms to the standard form gives

h = 8

and

k = -1

. Thus, the center of the circle is

(8, -1)

State the standard form of a circle's equation

The standard equation of a circle with center

(h, k)

and radius

r

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

This establishes the template to match the given equation against.

Rewrite the given equation to match the standard form exactly

The given equation

(x - 8)^2 + (y + 1)^2 = 81

can be rewritten as

(x - 8)^2 + (y - (-1))^2 = 9^2

Rewriting helps clearly identify the signs of

h

and

k

since the standard form uses subtraction:

(x - h)

and

(y - k)

Extract the center coordinates

(h, k)

h = 8

and

k = -1

, giving the coordinates

(8, -1)

By matching the rewritten equation to the standard form, we find the specific values of

h

and

k

that define the circle's center.

Identifying the center of a circle from its standard equation form

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