JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Area Under Curves

The area of the region bounded by the parabola

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

, the tangent to it at the point whose ordinate is 3 and the

x

-axis is:

Answer: (B)

Solution

The parabola

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

meets

y = 3

x = 2

, so the point of contact is

(2, 3)

. Differentiating,

2 (y - 2) d y = d x

, so the slope there is

\frac{1}{2}

and the tangent is

y = \frac{x}{2} + 2

, meeting the

x

-axis at

(- 4, 0)

. Integrating over

y

from

0

3

, the right boundary is the parabola

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

and the left boundary is the tangent

x = 2 y - 4

, so the width is

(y - 2)^{2} + 1 - (2 y - 4) = (y - 3)^{2}

and

Area = \int_{0}^{3} (y - 3)^{2} d y = [\frac{( y - 3 ) ^{3}}{3}]_{0}^{3} = 9.

The area of the region is

9

square units.

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Single correct option