What is the area, in square units, of the circle defined by the equation x^2 + y^2 + 8x - … — ACT

What is the area, in square units, of the circle defined by the equation $x^2 + y^2 + 8x - 10y + 5 = 0$ in the standard $(x, y)$ coordinate plane?

The area of the circle is

36\pi

square units.

To find the area of the circle, we first rewrite the given equation

x^2 + y^2 + 8x - 10y + 5 = 0

in standard form

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

. Rearranging the terms yields

(x^2 + 8x) + (y^2 - 10y) = -5

. Completing the square for

x

requires adding

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

to both sides, and completing the square for

y

requires adding

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

to both sides. This gives

(x^2 + 8x + 16) + (y^2 - 10y + 25) = -5 + 16 + 25

, which simplifies to

(x + 4)^2 + (y - 5)^2 = 36

. Thus, the squared radius is

r^2 = 36

. The area of the circle is

\pi r^2 = 36\pi

square units.

Group the

x

-terms and

y

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

(x^2 + 8x) + (y^2 - 10y) = -5

This prepares the equation for completing the square.

Complete the square for both the

x

and

y

terms by adding

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

and

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

to both sides.

(x^2 + 8x + 16) + (y^2 - 10y + 25) = -5 + 16 + 25

This allows the quadratic expressions to be written as perfect squares.

Simplify the equation into the standard form of a circle

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

(x + 4)^2 + (y - 5)^2 = 36

This standard form directly reveals that the radius squared (

r^2

) is

36

Calculate the area of the circle using the formula

A = \pi r^2

A = \pi (36) = 36\pi

The area of a circle is calculated by multiplying

\pi

by the square of the radius.

Converting the general equation of a circle to standard form by completing the square to determine its radius and calculate its area.

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