The equation of a circle is (x^{2} + y^{2} – 8x + 9y + 15 = 0). Find its radius.
-
A.
5 -
B.
(frac{1}{2}sqrt{15}) -
C.
(frac{1}{2}sqrt{85}) -
D.
(sqrt{85})
Correct Answer: Option C
Explanation
The equation of a circle is given as ((x – a)^{2} + (y – b)^{2} = r^{2}).
Expanding, we have: (x^{2} – 2ax + a^{2} + y^{2} – 2by + b^{2} = r^{2})
(x^{2} + y^{2} – 2ax – 2by + a^{2} + b^{2} = r^{2})
Comparing with the equation, (x^{2} + y^{2} – 8x + 9y = -15), we have
(2a = 8; 2b = -9; r^{2} – a^{2} – b^{2} = -15)
(a = 4; b = frac{-9}{2})
(therefore r^{2} = -15 + 4^{2} + (frac{-9}{2})^{2})
= (-15 + 16 + frac{81}{4} = frac{85}{4})
(r = sqrt{frac{85}{4} = frac{1}{2}sqrt{85})