Question

Using the following equation, find the center and radius of the circle. You must show all work and calculations to receive credit.
x2 − 4x + y2 + 8y = −4

Answer 1

Answer:

$r = 4$

The center is:

(2, -4)

Step-by-step explanation:

We have the following equation

$x^2 - 4x + y^2 + 8y = -4$

The general equation of a circle has the following form

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

Where r is the radius and the point (h, k) is the center of the circle

To transform the given equation in the general equation of a circumference we must use the method of square completions

Step 1

$(x^2 - 4x) + (y^2 + 8y) = -4$

Step 2

Add and subtract $(\frac{b}{2})^2$ For an equation of the form

$ax^2 +bx +c$

$(x^2 - 4x +4)-4 + (y^2 + 8y +16) -16= -4$

Step 3   Factor the expressions within the parentheses

$(x-2)^2-4 + (y+4)^2 -16= -4$

Step 4 Simplify

$(x-2)^2 + (y+4)^2= -4+16+4$

$(x-2)^2 + (y+4)^2= 4^2$

Finally

$r = 4$

The center is:

(2, -4)