JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Hyperbola

Let the tangent drawn to the parabola

y^{2} = 24 x

at the point

(α, β)

is perpendicular to the line

2 x + 2 y = 5

. Then the normal to the hyperbola

\frac{x ^{2}}{α ^{2}} - \frac{y ^{2}}{β ^{2}} = 1

at the point

(α + 4, β + 4)

does NOT pass through the point :

Answer: (D)

Solution

The tangent to the parabola

y^{2} = 24 x

(α, β)

has slope given by differentiating:

2 y \frac{d y}{d x} = 24 \Rightarrow \frac{d y}{d x} = \frac{12}{y} = \frac{12}{β}

. Since the tangent is perpendicular to

2 x + 2 y = 5

(slope

- 1

), we have

\frac{12}{β} = 1 \Rightarrow β = 12

. Since

(α, β)

lies on the parabola:

β^{2} = 24 α \Rightarrow 144 = 24 α \Rightarrow α = 6

. Thus

(α + 4, β + 4) = (10, 16)

. The normal to the hyperbola

\frac{x ^{2}}{36} - \frac{y ^{2}}{144} = 1

(10, 16)

is given by

\frac{36 x}{10} + \frac{144 y}{16} = 36 + 144 \Rightarrow 3.6 x + 9 y = 180 \Rightarrow 2 x + 5 y = 100

. Checking each option:

(15, 13)

gives

2 (15) + 5 (13) = 30 + 65 = 95 \neq = 100

, so the normal does NOT pass through

(15, 13)

JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Single correct option