Let \(\mathrm{y}=\mathrm{p}(\mathrm{x})\) be the parabola passing through the points \((-1,0),(0,1)\) and \((1,0)\). If the area of the region \(\left\{(x, y):(x+1)^{2}+(y-1)^{2} \leq 1, y \leq p(x)\right\}\) is A , then \(12(\pi-4 \mathrm{~A})\) is equal to ____.

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Area Under Curves

Let

y = p (x)

be the parabola passing through the points

(- 1, 0), (0, 1)

and

(1, 0)

. If the area of the region

{(x, y) : (x + 1)^{2} + (y - 1)^{2} \leq 1, y \leq p (x)}

is A , then

12 (π - 4 A)

is equal to ____.

Answer: 16

The parabola through

(- 1, 0)

(0, 1)

, and

(1, 0)

with vertical axis is

y = 1 - x^{2}

.

The region is bounded by the circle

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

(center

(- 1, 1)

, radius 1) and below the parabola.

The area is:

A = \int_{- 1}^{0} [(1 - x^{2}) - (1 - 1 - (x + 1)^{2})] d x = \int_{- 1}^{0} [- x^{2} + 1 - (x + 1)^{2}] d x

Evaluating:

A = [- \frac{x ^{3}}{3} + \frac{( x + 1 ) 1 - ( x + 1 ) ^{2}}{2} + \frac{1}{2} sin^{- 1} (x + 1)]_{- 1}^{0} = \frac{π}{4} - \frac{1}{3}

.

Then

12 (π - 4 A) = 12 (π - 4 (\frac{π}{4} - \frac{1}{3})) = 12 \cdot \frac{4}{3} = 16

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Integer answer