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

Correct Answer: 1575

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