Circles
Circle
Allen Star Batch
Grade 11
Question:
Let $S_1$ and $S_2$ denote the circles $x^2 + y^2 + 10x - 24y - 87 = 0$ and $x^2 + y^2 - 10x - 24y + 153 = 0$ respectively. (Let $m$ be the smallest positive value of $'a'$ for which the line $y = ax$ contains the centre of a circle which touches $S_2$ externally and $S_1$ internally). Given that $m^2 = \frac{p}{q}$, where $p$ and $q$ are relatively prime integers, if $(p + q)$ is equal to $13^k$, then the value of $k$ is equal to ______.
Step-by-Step Solution
Key Concept: A circle tangent to two given circles has its center locus on a conic section; use the sum/difference of distances criterion to identify the conic.
Circle $S_1$ has center $C_1(-5, 12)$ and radius $r_1 = 16$; circle $S_2$ has center $C_2(5, 12)$ and radius $r_2 = 4$. For a circle $C$ touching $S_1$ internally and $S_2$ externally: $CC_1 + CC_2 = 16 - r + r + 4 = 20$. Thus $C$ lies on an ellipse with foci at $C_1, C_2$ and major axis length $2a = 20$. The ellipse equation is $\frac{x^2}{100} + \frac{(y-12)^2}{75} = 1$. For tangent $y = mx$ from origin: $-12 = m\sqrt{100m^2 + 75}$ gives $m^2 = \frac{69}{100}$, so $p + q = 169$ and $k = 2$.
Correct Answer: 2