JEE Main · 2022 · 25 Jul 2022 · Shift 1 · Parabola

Let the locus of the centre

(α, β), β > 0

, of the circle which touches the circle

x^{2} + (y - 1)^{2} = 1

externally and also touches the x-axis be L. Then the area bounded by L and the line

y = 4

is:

Answer: (C)

Solution

Let the variable circle have center

(α, β)

with

β > 0

and radius

r

. Since it touches the x-axis,

r = β

. Since it touches the given circle

x^{2} + (y - 1)^{2} = 1

(center

(0, 1)

, radius

1

) externally, the distance between centers equals the sum of radii:

α^{2} + (β - 1)^{2} = β + 1

. Squaring:

α^{2} + (β - 1)^{2} = (β + 1)^{2}

, which simplifies to

α^{2} = 4 β

. The locus L is the parabola

x^{2} = 4 y

. The area bounded by this parabola and the line

y = 4

A = 2 \int_{0}^{4} (4 - \frac{x ^{2}}{4}) d x = 2 [4 x - \frac{x ^{3}}{12}]_{0}^{4} = 2 (16 - \frac{16}{3}) = \frac{64}{3}

JEE Main · 2022 · 25 Jul 2022 · Shift 1 · Single correct option