Circles
Circle
Allen Star Batch
Grade 11
Question:
The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance $6$ unit from each other. Let $P$ be the mid-point of the line segment joining the centres of $C_1$ and $C_2$. If a common tangent to $C_1$ and $C_2$ passing through $P$ is also a common tangent to $C_2$ and $C$, then the radius of the circle $C$ is ___.
Step-by-Step Solution
Key Concept: The tangent of the angle from the perpendicular can be expressed in two ways using the geometric configuration and trigonometric relations.
From the given conditions $PB = 3$, $BN = 1$, triangle $ABO$ is isosceles. Since $\tan\alpha = \frac{1}{2\sqrt{2}}$ and $OP$ is perpendicular to $AB$, we have $\tan(\angle AOP) = \frac{2\sqrt{2}}{R}$. Equating these expressions: $\frac{1}{2\sqrt{2}} = \frac{2\sqrt{2}}{R}$ yields $R = 8$.
Correct Answer: 8