A circle in the standard (x, y) coordinate plane is defined by the equation x^2 + y^2 - 4x… — ACT

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $x^2 + y^2 - 4x + 6y - 12 = 0$ . What is the radius of this circle?

The radius of the circle is 5.

The correct answer is 5. Grouping the

x

-terms and

y

-terms of the equation

x^2 + y^2 - 4x + 6y - 12 = 0

gives

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

. Completing the square requires adding

(\frac{-4}{2})^2 = 4

and

(\frac{6}{2})^2 = 9

to both sides, yielding

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

, which simplifies to

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

. In the standard circle equation form

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

r^2 = 25

, meaning the radius

r

\sqrt{25} = 5

Group the

x

-terms and

y

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

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

This prepares the quadratic expression to be written in standard circle form by completing the square.

Find the constant values needed to complete the square for both the

x

and

y

variables. Add these constants to both sides of the equation.

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

The constant for

x

(\frac{-4}{2})^2 = 4

, and the constant for

y

(\frac{6}{2})^2 = 9

. Adding them to both sides maintains equality.

Factor the perfect square trinomials on the left side and simplify the right side.

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

This puts the equation in the standard form

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

Identify

r^2

from the standard form and solve for the radius

r

r = \sqrt{25} = 5

Taking the square root of the constant on the right side yields the radius of the circle.

Rewriting a circle's equation from general form to standard form by completing the square to find its radius.

For a circle given in the general form

x^2 + y^2 + Dx + Ey + F = 0

, the radius can be calculated directly using the formula

r = \frac{1}{2}\sqrt{D^2 + E^2 - 4F}

. Substituting the coefficients

D = -4

E = 6

, and

F = -12

gives:

r = \frac{1}{2}\sqrt{(-4)^2 + 6^2 - 4(-12)} = \frac{1}{2}\sqrt{16 + 36 + 48} = \frac{1}{2}\sqrt{100} = \frac{1}{2}(10) = 5

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