Question

If the line \( 5x - 2y - 6 = 0 \) is a tangent to the hyperbola \( 5x^2 - ky^2 = 12 \), then the equation of the normal to this hyperbola at \( (\sqrt{6}, p) \) is:

• A. \( \sqrt{6}x + 2y = 0 \)
• B. \( 2\sqrt{6}x + 3y = 3 \)
• C. \( \sqrt{6}x - 5y = 21 \)
• D. \( 3\sqrt{6}x - y = 21 \)

Hint:

For hyperbola tangents and normals, differentiate implicitly and substitute known values carefully.

Answer 1

The Correct Option is C

Step 1: Finding the normal equation Using differentiation for the hyperbola, \[ \frac{dy}{dx} = \frac{5x}{ky} \] Substituting \( x = \sqrt{6} \), solving for \( y \), and forming the normal equation gives: \[ \sqrt{6}x - 5y = 21 \]