JEE Advanced · 2020 · 27 Sep 2020 · Paper 2 · Hyperbola

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

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?

Answer: (A), (D)

Solution

Since the normal makes equal intercepts, its slope is

- 1

, so the tangent has slope

1

. The tangent equation is

y = x - 1

. For a tangent at

(x_{1}, y_{1})

\frac{x x _{1}}{a ^{2}} - \frac{y y _{1}}{b ^{2}} = 1

. Comparing gives

x_{1} = a^{2}

y_{1} = b^{2}

, and

a^{2} - b^{2} = 1

. The eccentricity

e = 1 + \frac{b ^{2}}{a ^{2}} = 1 + \frac{b ^{2}}{b ^{2} + 1}

. Since

\frac{b ^{2}}{b ^{2} + 1} < 1

, we have

1 < e < 2

(option A). The normal equation is

x + y = a^{2} + b^{2}

. The triangle area

Δ = \frac{1}{2} (a^{2} + b^{2} - 1) b^{2} = \frac{1}{2} (2 b^{2}) b^{2} = b^{4}

(option D).

JEE Advanced · 2020 · 27 Sep 2020 · Paper 2 · One or more correct options