Answer:
First we must use the property of the sum of perfect squares:
X^2 -14x +y^2 +10y+65 =0
Adding 7^2 and -7^2
x^2 -14x+7^2-7^2 +y^2+10y+65=0
x^2 -2.7x +7^2 -49 +y^2+10y+65=0
(X-7)^2 +y^2+10y+16=0
Adding 5^2 and -5^2
(X-7)^2 +y^2 +10y+5^2-5^2+16=0
(X-7)^2 +y^2+2.5y+5^2 -25 +16=0
(X-7)^2+ (y+5)^2 -9 = 0
(X-7)^2 +(y+5)^2 = 9
(X-7)^2+(y+5)^2 = 3^2
Center = (7, -5)
R = 3
The circle is in the 4 quadrant.
