In the xy-plane, a circle is defined by the equation x^2 + y^2 + 6x - 8y = 0. Which of the… — SAT

In the $xy$ -plane, a circle is defined by the equation $x^2 + y^2 + 6x - 8y = 0$ . Which of the following points lies on the circle?

The point

(2, 4)

lies on the circle.

Substituting the coordinates of the point

(2, 4)

into the circle's equation

x^2 + y^2 + 6x - 8y = 0

yields

2^2 + 4^2 + 6(2) - 8(4) = 4 + 16 + 12 - 32 = 0

, which is a true statement. Therefore, this point lies on the circle.

Group the

x

and

y

terms and complete the square for both variables.

The expression

(x^2 + 6x) + (y^2 - 8y) = 0

becomes

(x^2 + 6x + 9) + (y^2 - 8y + 16) = 9 + 16

, which simplifies to

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

Converting the general form equation of a circle into the standard form

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

helps identify its center and radius.

Determine the center and the radius from the standard form equation.

The center is

(h, k) = (-3, 4)

and the radius is

r = \sqrt{25} = 5

Understanding the circle's parameters allows us to analyze which points lie on its perimeter.

Substitute the given options into the standard form equation to verify which point lies on the circle.

For

(2, 4)

, we get

(2 + 3)^2 + (4 - 4)^2 = 5^2 + 0^2 = 25

, which is true. For the other points, the equations are not satisfied.

A point lies on a circle if its distance to the center is exactly equal to the radius, meaning its coordinates satisfy the circle's equation.

Equations of Circles in the Coordinate Plane

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