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 the common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha \beta)^2$ equals ____.

Answer 1

Identify the center and radius of each circle:

- $C_1 : x^2 + y^2 = 25$ has center $(0, 0)$ and radius $R_1 = 5$.

- $C_2 : (x - \alpha)^2 + y^2 = 16$ has center $(\alpha, 0)$ and radius $R_2 = 4$.

Distance between centers:

$d = \alpha$

Length of the common chord:

$\beta = 2 \sqrt{5^2 - \left( \frac{\alpha^2 + 9}{2\alpha} \right)^2} = 2 \sqrt{25 - \left( \frac{\alpha^2 + 9}{2\alpha} \right)^2}.$

Using $\sin \theta = \frac{\sqrt{63}}{8}$, calculate $\alpha \beta$:

$\alpha \beta = 5 \sqrt{63}$

Final calculation : $(\alpha \beta)^2 = 25 \times 63 =1575$