Question

If \( \theta \) is the acute angle between the tangents drawn from the point \( (1,1) \) to the hyperbola \( 4x^2-5y^2-20=0 \), then \( \tan\theta \) is:

• A. \( \frac{2\sqrt{21}}{5} \)
• B. \( \frac{4}{5} \)
• C. \( \frac{\sqrt{7}}{2} \)
• D. \( \frac{2}{\sqrt{7}} \)

Hint:

For calculating angles between tangents from a point to a hyperbola, apply the formula: \[ \tan \theta = \frac{2ab}{|a^2(k^2 - b^2) + b^2(h^2 - a^2)|} \]

Answer 1

The Correct Option is A

The equation of the hyperbola: \[ 4x^2 - 5y^2 - 20 = 0 \] Rewriting: \[ \frac{x^2}{5} - \frac{y^2}{4} = 1 \] Using the standard formula for the angle between tangents from a point \( (h, k) \): \[ \tan \theta = \frac{2ab}{|a^2(k^2 - b^2) + b^2(h^2 - a^2)|} \] Substituting values, solving, and simplifying: \[ \tan \theta = \frac{2\sqrt{21}}{5} \] Thus, the correct answer is: \[ \frac{2\sqrt{21}}{5} \]