Circles
Circle
Allen Star Batch
Grade 11
Question:
One circle has a radius of 5 units and its center is at (0, 5). A second circle has a radius of 12 and its centre is at (12, 0). The radius of a third circle which passes through the center of the second circle and both points of intersection of the first two circles, is equal to:
13/2
15/2
17/2
None of these
Step-by-Step Solution
Key Concept: Using the family of circles through the intersection of two given circles allows parameter determination via a third point.
The required circle is given by $(x^2 + y^2 - 10y) + k(x^2 + y^2 - 24x) = 0$. Since it passes through $(12, 0)$: $144 + k(144 - 288) = 0$, giving $k = 1$. The equation becomes $x^2 + y^2 - 12x - 5y = 0$ with radius $\frac{13}{2}$.
Correct Answer: 1