Question

If \( \theta \) is the acute angle between the tangents drawn from the point \( (2, 3) \) to the hyperbola \( 5x^2 - 6y^2 - 30 = 0 \), then \( \tan \theta \) is:

• A. \( \frac{\pi}{4} \)
• B. \( \frac{3}{4} \)
• C. \( \frac{4}{3} \)
• D. \( \frac{\pi}{2} \)

Hint:

For the acute angle between tangents, always use the standard formula for conic sections to solve. Make sure to check the signs and simplify the terms.

Answer 1

The Correct Option is C

The equation of the hyperbola is given by: \[ 5x^2 - 6y^2 - 30 = 0 \] We can rewrite the equation as: \[ \frac{x^2}{6} - \frac{y^2}{5} = 1 \] This represents the standard form of the hyperbola. The formula for the angle between two tangents drawn from an external point to a conic is: \[ \tan \theta = \frac{2\sqrt{AB}}{|A + B|} \] where \( A = 5 \), \( B = 6 \) (from the equation of the hyperbola). Thus, \[ \tan \theta = \frac{2\sqrt{5 \times 6}}{|5 + 6|} = \frac{2\sqrt{30}}{11} \] After simplifying, we get \( \tan \theta = \frac{4}{3} \).