If the area of the region bounded by the parabola y 2 = 4 k x and the line x = k , (where k 0 ) is 3 128 sq. units, then the value of sin − 1 ( k 2 ) is equal to...

Question

If the area of the region bounded by the parabola y 2 = 4 k x and the line x = k , (where k > 0 ) is 3 128 ​ sq. units, then the value of sin − 1 ( k 2 ​ ) is equal to...

TestHub · Accepted Answer

The area of the region bounded by the parabola $y^{2} = 4 k x$ and the line $x = k$ is given by

$A = 2 \int_{0}^{k} y d x$

$A = 2 \int_{0}^{k} 4 k x d x$

$A = 4 k [\frac{2}{3} x^{3/2}]_{0}^{k}$

$A = \frac{8}{3} k^{2}$

Given that the area is $\frac{128}{3}$ , we have

$\frac{8}{3} k^{2} = \frac{128}{3}$

$k^{2} = 16$

Since $k > 0$ , we get $k = 4$ .

Substituting $k = 4$ :

$sin^{- 1} (\frac{2}{k}) = sin^{- 1} (\frac{2}{4}) = sin^{- 1} (\frac{1}{2}) = \frac{π}{6}$

Answer: $\frac{π}{6}$

Mathematics - Area Under Curves Question with Solution | TestHub

Options:

Doubts & Discussion