Question

Given the equation x^2+y^2+4x+16y-15=0, (a) find the center and radius; (b) find the intercepts, if any. Please show all of your work.

Answer 1

First, get add 15 to both sides:
(x^2 + y^2 + 4x + 16y -15) +15 = (0) + 15
x^2 + y^2 + 4x + 16y = 15

Second, complete the square for both x and y. In order to do this, take the coefficient of the first degree x or y value, halve it, square the half, and add this to both sides:
(x^2 + y^2 + 4x + 16y) + (4/2)^2 + (16/2)^2 = (15) + (4/2)^2 + (16/2)^2
(x^2 + y^2 + 4x + 16y) + 4 + 64 = (15) + 4 + 64
x^2 + y^2 + 4x +16y + 4 + 64 = 85

Next, regroup the values on the variable side so that all x and y values are grouped:
(x^2 + 4x + 4) + (y^2 + 16y +64) = 85

Simplify this to get the equation of a circle in standard form:
(x +2)^2 + (y + 8)^2 = 85

Using the equation (x - h)^2 + (y - k)^2 = r^2, find the center (h, k) and the radius (r):
Center at (-2, -8) with radius of (square root of 85)