JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · Parabola

Consider the parabola

y^{2} = 4 x

. Let

S

be the focus of the parabola. A pair of tangents drawn to the parabola from the point

P = (- 2, 1)

meet the parabola at

P_{1}

and

P_{2}

. Let

Q_{1}

and

Q_{2}

be points on the lines

S P_{1}

and

S P_{2}

respectively such that

P Q_{1}

is perpendicular to

S P_{1}

and

P Q_{2}

is perpendicular to

S P_{2}

. Then, which of the following is/are TRUE ?

Answer: (B), (C), (D)

Solution

For parabola

y^{2} = 4 x

, focus

S = (1, 0)

. Tangent from

P (- 2, 1)

with slope

m

y = m x + \frac{1}{m}

.

Passing through

(- 2, 1)

1 = - 2 m + \frac{1}{m}

, giving

m^{2} + 2 m - 1 = 0

, so

m = - 1

m = \frac{1}{2}

.

Points of tangency:

P_{1} (4, 4)

(for

m = \frac{1}{2}

) and

P_{2} (1, - 2)

(for

m = - 1

).

Line

S P_{1}

\frac{x - 1}{3} = \frac{y}{4}

, or

4 x - 3 y - 4 = 0

.
Line

S P_{2}

x = 1

.

For

Q_{1}

S P_{1}

with

P Q_{1} ⊥ S P_{1}

P Q_{1} = \frac{4 ( - 2 ) - 3 ( 1 ) - 4}{5} = \frac{15}{5} = 3

✓

S P = 9 + 1 = 10

. In right triangle

S P Q_{1}

S Q_{1} = 10 - 9 = 1

✓

Similarly,

S Q_{2} = 1

✓ and by geometry,

Q_{1} Q_{2} = \frac{3 10}{5}

✓

Options (B), (C), (D) are correct.

JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · One or more correct options