Circles
Common Chord / Area of Triangle
nta_pyq_2024_jan
Grade 11
Question:
Consider two circles $C_1: x^2+y^2=25$ and $C_2:(x-\alpha)^2+y^2=16$, where $\alpha\in(5,9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha\beta)^2$ equals
Step-by-Step Solution
Key Concept: Let $P$ be intersection point. Area of $\triangle OAP$ (where $O$ is origin, $A=(\alpha,0)$): use $\frac{1}{2}\cdot5\cdot4\cdot\sin\theta$ where $\sin\theta=\frac{\sqrt{63}}{8}$. Also area $=\frac{1}{2}\cdot\alpha\cdot\frac{\beta}{2}$. Equate to find $\alpha\beta$.
Area of $\triangle OAP=\frac{1}{2}\cdot5\cdot4\cdot\sin\theta=10\cdot\frac{\sqrt{63}}{8}=\frac{5\sqrt{63}}{4}$. Also $=\frac{1}{2}\cdot\alpha\cdot\frac{\beta}{2}=\frac{\alpha\beta}{4}$. So $\alpha\beta=5\sqrt{63}$, $(\alpha\beta)^2=25\times63=1575$.
Correct Answer: 1575