Let the line \(\mathrm{L}: \sqrt{2} \mathrm{x}+\mathrm{y}=\alpha\) pass through the point of the intersection P (in the first quadrant) of the circle \(x^{2}+y^{2}=3\) and the parabola \(x^{2}=2 y\). Let the line L touch two circles \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\) of equal radius \(2 \sqrt{3}\). If the centres \(\mathrm{Q}_{1}\) and \(\mathrm{Q}_{2}\) of the circles \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\) lie on the \(y\)-axis, then the square of the area of the triangle \(\mathrm{PQ}_{1} \mathrm{Q}_{2}\) is equal to ______.

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Circle

Let the line

L : 2 x + y = α

pass through the point of the intersection P (in the first quadrant) of the circle

x^{2} + y^{2} = 3

and the parabola

x^{2} = 2 y

. Let the line L touch two circles

C_{1}

and

C_{2}

of equal radius

23

. If the centres

Q_{1}

and

Q_{2}

of the circles

C_{1}

and

C_{2}

lie on the

y

-axis, then the square of the area of the triangle

PQ_{1} Q_{2}

is equal to ______.

Answer: 72

Circle

x^{2} + y^{2} = 3

meets parabola

x^{2} = 2 y

where

2 y + y^{2} = 3 \Rightarrow (y + 3) (y - 1) = 0

. Taking

y = 1

(first quadrant):

x = 2

, so

P = (2, 1)

P

lies on

L : 2 x + y = α \Rightarrow 2 + 1 = α \Rightarrow α = 3

, so

L : 2 x + y = 3

.
Let

Q = (0, k)

on the

y

-axis be the centre of a circle of radius

23

tangent to

L

\frac{∣0 + k - 3∣}{2 + 1} = 23 ⟹ ∣ k - 3∣ = 6 ⟹ k = 9 or k = - 3

Q_{1} = (0, 9), Q_{2} = (0, - 3)

. Area of

△ P Q_{1} Q_{2}

with base

Q_{1} Q_{2}

on the

y

-axis (length 12) and horizontal distance from

P

to the axis

= 2

Area = \frac{1}{2} \times 12 \times 2 = 62 ⟹ (Area)^{2} = 72

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Integer answer