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 x²/a² - y²/b² = 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 Δ 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 < √2 

(B) √2 < e < 2 

(C) Δ = a⁴ 

(D) Δ = b⁴

Answer 1

(A) 1 < e < √2, (D) Δ = b⁴

Since Normal at point P makes equal intercept on co-ordinate axes, therefore slope of Normal = –1 

Hence slope of tangent = 1 

Equation of tangent 

y – 0 = 1(x–1) 

y = x – 1 

Equation of tangent at (x₁y₁)

Answer 2

Area of triangle is b⁴.

eccentricity lies between 1<e<√2.

option: A and D is correct✅

Answer 3

option A and D is correct