Question

The length of the transverse axis of a hyperbola is \( 2\cos \alpha \). The foci of the hyperbola are the same as that of the ellipse \( 9x^2 + 16y^2 = 144 \). The equation of the hyperbola is:

• A. \( \frac{x^2}{\cos^2 \alpha} - \frac{y^2}{7 - \cos^2 \alpha} = 1 \)
• B. \( \frac{x^2}{\cos^2 \alpha} - \frac{y^2}{7 + \cos^2 \alpha} = 1 \)
• C. \( \frac{x^2}{1 + \cos^2 \alpha} - \frac{y^2}{7 - \cos^2 \alpha} = 1 \)
• D. \( \frac{x^2}{1 + \cos^2 \alpha} - \frac{y^2}{7 + \cos^2 \alpha} = 1 \)
• E. \( \frac{x^2}{\cos^2 \alpha} - \frac{y^2}{5 - \cos^2 \alpha} = 1 \)

Hint:

In confocal problems, always calculate $c^2$ (the square of the distance from center to focus) first. It acts as the "bridge" between the two different conic equations.

Answer 1

The Correct Option is A