Question

The tangent drawn at an extremity (in the first quadrant) of latus rectum of the hyperbola $$ \frac{x^2}{4} - \frac{y^2}{5} = 1 $$ meets the x-axis and y-axis at $ A $ and $ B $ respectively. If $ O $ is the origin, find $$ (OA)^2 - (OB)^2. $$

• A. \( \frac{20}{9} \)
• B. \( \frac{16}{9} \)
• C. \( -\frac{4}{9} \)
• D. \( \frac{4}{3} \)

Hint:

Use hyperbola and tangent properties along with intercept form for calculations.

Answer 1

The Correct Option is A

1. Find coordinates of latus rectum extremity.
2. Equation of tangent at this point.
3. Calculate intercepts \( OA \) and \( OB \).
4. Compute \( (OA)^2 - (OB)^2 = \frac{20}{9} \).