The equation of a circle is \(3x^{2} + 3y^{2} + 24x - 12y = 15\). Find its radius.
Question 1 Report
The equation of a circle is \(3x^{2} + 3y^{2} + 24x - 12y = 15\). Find its radius.
Answer Details
To find the radius of the circle, we need to use the standard form of the equation of a circle, which is \((x - a)^{2} + (y - b)^{2} = r^{2}\), where \((a, b)\) is the center of the circle and \(r\) is the radius.
To convert the given equation to standard form, we can complete the square for both \(x\) and \(y\):
\begin{align*}
3x^{2} + 3y^{2} + 24x - 12y &= 15 \\
3(x^{2} + 8x) + 3(y^{2} - 4y) &= 15 \\
3(x^{2} + 8x + 16) + 3(y^{2} - 4y + 4) &= 15 + 3(16) + 3(4) \\
3(x + 4)^{2} + 3(y - 2)^{2} &= 72 \\
(x + 4)^{2} + (y - 2)^{2} &= 8^{2}
\end{align*}
Comparing this with the standard form, we see that the center of the circle is \((-4, 2)\) and the radius is \(8\). Therefore, the answer is (d) 5.
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