Question:
The ratio of the areas of the concentric circles \(x^2 + y^2 - 6x + 12y + 15 = 0\) and \(x^2 + y^2 - 6x + 12y - 15 = 0\) is
The ratio of the areas of the concentric circles \(x^2 + y^2 - 6x + 12y + 15 = 0\) and \(x^2 + y^2 - 6x + 12y - 15 = 0\) is
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For concentric circles, complete the square to find radii, then ratio of areas is ratio of squares of radii.
Updated On: Jul 18, 2026
- \(1:\sqrt{2}\)
- \(1:\sqrt{3}\)
- \(1:2\)
- \(1:4\)
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The Correct Option is C
Solution and Explanation
Step 1: Find centers and radii.
Circle 1: \(x^2 + y^2 - 6x + 12y + 15 = 0\)
Complete square: \((x-3)^2 + (y+6)^2 = 0\) ???
Let's correct: \((x^2 - 6x + 9) + (y^2 + 12y + 36) = -15 + 9 + 36 = 30\)
\(\Rightarrow (x-3)^2 + (y+6)^2 = 30\), radius \(r_1 = \sqrt{30}\)
Step 2: Second circle.
\(x^2 + y^2 - 6x + 12y - 15 = 0\)
Complete square: \((x-3)^2 + (y+6)^2 = 60\), radius \(r_2 = \sqrt{60}\)
Step 3: Area formula.
\(\text{Area} = \pi r^2\)
\(\text{Area ratio} = \frac{\pi r_1^2}{\pi r_2^2} = \frac{30}{60} = \frac{1}{2}\)
Step 4: Final conclusion.
Hence, the ratio of areas is \[ \boxed{1:2} \]
Circle 1: \(x^2 + y^2 - 6x + 12y + 15 = 0\)
Complete square: \((x-3)^2 + (y+6)^2 = 0\) ???
Let's correct: \((x^2 - 6x + 9) + (y^2 + 12y + 36) = -15 + 9 + 36 = 30\)
\(\Rightarrow (x-3)^2 + (y+6)^2 = 30\), radius \(r_1 = \sqrt{30}\)
Step 2: Second circle.
\(x^2 + y^2 - 6x + 12y - 15 = 0\)
Complete square: \((x-3)^2 + (y+6)^2 = 60\), radius \(r_2 = \sqrt{60}\)
Step 3: Area formula.
\(\text{Area} = \pi r^2\)
\(\text{Area ratio} = \frac{\pi r_1^2}{\pi r_2^2} = \frac{30}{60} = \frac{1}{2}\)
Step 4: Final conclusion.
Hence, the ratio of areas is \[ \boxed{1:2} \]
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