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

Answer: (A), (D)

$1 < e < \sqrt{2}$

Answer 2

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