Question

For the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the locus of the point of intersection of the tangents at the endpoints of normal chords is:

• A. \( \frac{a^6}{x^2} + \frac{b^6}{y^2} = (a^2 + b^2)^2 \)
• B. \( \frac{a^6}{x^2} + \frac{b^6}{y^2} = (a^2 + b^2)^2 \)
• C. \( \frac{a^6}{x^2} - \frac{b^6}{y^2} = (a^2 - b^2)^2 \)
• D. \( \frac{a^6}{x^2} + \frac{b^6}{y^2} = (a^2 - b^2)^2 \)

Hint:

For hyperbolas, use standard identities and known results involving normals and tangents to quickly identify loci of special points.

Answer 1

The Correct Option is B

This is a standard result in conics. For the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the normal chords are those perpendicular to the curve.
The tangents at endpoints of these chords intersect at points whose locus is given by: \[ \frac{a^6}{x^2} + \frac{b^6}{y^2} = (a^2 + b^2)^2 \] This is derived from the parametric coordinates of points on the hyperbola and analyzing the envelope of normal chords.