Decide whether or not the equation has a circle as its graph

Decide whether or not the equation has a circle as its graph. If it does, give the center and the radius. If it does not. describe the graph. X^2 + y^2 - 10x + 6y = -18 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choic The graph of the equation is a circle with center. (Type an ordered pair.) The radius of the circle is .The graph of the equation is a line. The graph of the equation is a point. The graph is nonexistent.

Solution

We have

x² + y² -10x + 6y = -18

we will group x and y terms and make them perfect square.

x² - 10x + y² + 6y = -18

for making x terms as a perfect square we will divide the coefficient of \"x\" by 2 and square it then add it on both sides and similarly for the y-terms

x² - 10x + 25 + y² + 6y + 9 = -18 + 25 + 9                   // 10/2 = 5 and square of 5 = 25 , 6/2 = 3 and 3² = 9

(x - 5)² + (y + 3)² = 16

we can rewrite it as

(x - 5)² + (y - (-3))² = 4²

On comparing it with (x - h)² + (y - k)² = r²                   // (h,k) is the center of the circle and \"r\" is the radius

on comparing we get

h = 5, k = -3 and r = 4

center is (5, -3) and radius = 4