JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Parabola

Let

A (0, 1)

B (1, 1)

and

C (1, 0)

be the mid-points of the sides of a triangle with incentre at the point D. If the focus of the parabola

y^{2} = 4 a x

passing through D is

(α + β 2, 0)

, where

α

and

β

are rational numbers, then

\frac{α}{β ^{2}}

is equal to

Answer: (B)

Solution

A, B, C

are the mid-points of the sides, the vertices are

A + B - C = (0, 2)

B + C - A = (2, 0)

and

C + A - B = (0, 0)

. Label them

U (0, 2), V (2, 0), O (0, 0)

; the sides opposite them have lengths

∣ O V ∣ = 2, ∣ O U ∣ = 2, ∣ U V ∣ = 22

. The incentre uses these side lengths as weights:

D = \frac{2 ( 0 , 2 ) + 2 ( 2 , 0 ) + 2 2 ( 0 , 0 )}{2 + 2 + 2 2} = (\frac{4}{4 + 2 2}, \frac{4}{4 + 2 2}) = (2 - 2, 2 - 2),

which indeed lies on the axis of symmetry

y = x

. Since

D

lies on

y^{2} = 4 a x

(2 - 2)^{2} = 4 a (2 - 2) \Rightarrow 4 a = 2 - 2 \Rightarrow a = \frac{1}{2} - \frac{2}{4} .

The focus is

(a, 0)

, so

α = \frac{1}{2}

and

β = - \frac{1}{4}

, and

\frac{α}{β ^{2}} = \frac{1/2}{1/16} = 8.

JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Single correct option