Question

Find the locus of the point of intersection of tangents drawn at the ends of a **normal chord** of the hyperbola: $$ x^2 - y^2 = a^2 $$

• A. \( y^4 - x^4 = 4a^2 x^2 y^2 \)
• B. \( y^2 - x^2 = 4a^2 x^2 y^2 \)
• C. \( a^2 (y^2 - x^2) = 4x^2 y^2 \)
• D. \( y^2 + x^2 = 4a^2 x^2 y^2 \)

Hint:

This result is standard in hyperbola geometry for chords normal to the curve. Remember the form of tangents and substitute.

Answer 1

The Correct Option is C

Let ends of the normal chord be \( P \) and \( Q \). The intersection of tangents at \( P \) and \( Q \) is known to trace a locus. The general form of tangent to hyperbola \( x^2 - y^2 = a^2 \) at point \( (x_1, y_1) \) is: \[ x x_1 - y y_1 = a^2 \] Let the ends of the normal chord be parameterized by: \[ x = a \sec \theta,\quad y = a \tan \theta \] Then the tangent at this point is: \[ x \sec \theta - y \tan \theta = a \] Similarly, at \( \theta' \), you get: \[ x \sec \theta' - y \tan \theta' = a \] If \( PQ \) is a normal chord, the foot of perpendicular from center lies on it, and midpoint of PQ lies on conjugate hyperbola. Using advanced geometric result: the locus of intersection of tangents at ends of normal chord is: \[ a^2(y^2 - x^2) = 4x^2 y^2 \Rightarrow \boxed{ a^2(y^2 - x^2) = 4x^2 y^2 } \]