Question

Let $a$ and $b$ be positive real numbers such that \(a >1\) and \(b < a\). Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Suppose the tangent to the hyperbola at P passes through the point $(1,0)$, and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the $x$-axis. If e denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

• A. \(1 < e < \sqrt{2}\)
• B. \(\sqrt{2} < e < 2\)
• C. $\Delta= a ^{4}$
• D. $\Delta=b^{4}$

Answer 1

The Correct Option is A, D

(A) \(1 < e < \sqrt{2}\)
(D) $\Delta=b^{4}$