Let the eccentricity of the hyperbola H:x²/a² - y²/b² = 1 be √(5/2) and length of its latus rectum be 6√2. If y = 2x + c is a tangent to the hyperbola H, then the value of c² is equal to :

Solution

Maths sir · 4 steps

Get a² and b² from e and the latus rectum, then use the tangency condition c² = a²m² - b².

1. Use eccentricity. For this hyperbola e² = 1 + b²/a². With e² = 5/2 this gives b²/a² = 3/2, so b² = 3/2a².
2. Use latus rectum. The latus rectum is 2b²/a = 2/a × 3/2a² = 3a. Setting 3a = 6√2 gives a = 2√2, so a² = 8 and b² = 12.

   c² = a²m² - b²

3. Tangency condition. The line y = mx + c touches x²/a² - y²/b² = 1 exactly when the condition above holds. Here m = 2, so c² = 8 × 4 - 12.
4. So. c² = 32 - 12 = 20, which is option (b).

The idea to keep. For a tangent y = mx + c, the hyperbola needs c² = a²m² - b² and the ellipse needs c² = a²m² + b². Learn both side by side and watch the sign.

Answer from the NTA final answer key. Solution by PrepWiser.