Circles
Circle
Allen Star Batch
Grade 11
Question:
Let $C_1$ and $C_2$ be centres of two circles whose radii are $2$ and $4$ respectively. Also $C_1C_2 = 10$ and direct common tangents of these circles touch them at $P, Q, R, S$. Another circle of radius $\lambda$ is drawn passing through $P, Q, R, S$. Then
Midpoint of $C_1C_2$ is centre of the circle passing through $P, Q, R, S$
Centre of the circle passing through $P, Q, R, S$ divides $C_1C_2$ in the ratio $1:2$
$\lambda^2 = 33$
$\lambda^2 = 35$
Step-by-Step Solution
Key Concept: The center of the required circle lies on the radical axis and satisfies equidistance from multiple points on both given circles.
Let $T$ be the midpoint of $PR$, with perpendicular from $T$ to $PR$ meeting $C_1C_2$ at $M$. Using the condition $MP^2 = MR^2 = MQ^2 = MS^2 = MT^2 + TR^2$, we verify $M$ is the center of the required circle. Calculate $PR = \sqrt{(C_1C_2)^2 - (n_1 - n_2)^2} = \sqrt{100-4} = \sqrt{96}$. Then $\lambda^2 = MT^2 + TR^2 = \left(\frac{2+4}{2}\right)^2 + TR^2 = 9 + \frac{96}{4} = 9 + 24 = 33$.
Correct Answer: 1,3