Question

The tangents drawn to the hyperbola \(5x^2 - 9y^2 = 90\) through a variable point \(P\) make angles \(\alpha\) and \(\beta\) with its transverse axis. If \(\alpha\) and \(\beta\) are complementary angles, then the locus of \(P\) is

• A. \(x^2 + y^2 = 8\)
• B. \(x^2 - y^2 = 8\)
• C. \(x^2 - y^2 = 28\)
• D. \(x^2 + y^2 = 28\)

Hint:

For tangents forming complementary angles, use the identity \(\tan \alpha \cdot \tan \beta = 1\).

Answer 1

The Correct Option is C

Given: \(\alpha + \beta = \dfrac{\pi}{2} \Rightarrow \tan \alpha \cdot \tan \beta = 1\).
Use the condition for angle between tangents drawn from a point \(P(x, y)\) to the hyperbola. The tangents form complementary angles with the transverse axis if the product of their slopes equals 1, which leads to the condition: \(x^2 - y^2 = 28\).