The equation of a circle is \(x^{2} + y^{2} – 8x + 9y + 15 = 0\). Find its radius.
- 5
- \(\frac{1}{2}\sqrt{15}\)
-
\(\frac{1}{2}\sqrt{85}\)
- \(\sqrt{85}\)
The correct answer is: 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
\(2a = 8; 2b = -9; r^{2} - a^{2} - b^{2} = -15\)
\(a = 4; b = \frac{-9}{2}\)
\(\therefore r^{2} = -15 + 4^{2} + (\frac{-9}{2})^{2}\)
= \(-15 + 16 + \frac{81}{4} = \frac{85}{4}\)
\(r = \sqrt{\frac{85}{4} = \frac{1}{2}\sqrt{85}\)