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

Step-by-step solution

The area of the region bounded by the parabola y^2 = 4kx and the line x = k is given by A = 2 ₀^ k y , dx A = 2 ₀^ k 4kx , dx A = 4 k [ 2 3 x^ 3/2 ]₀^ k A = 8 3 k^2 Given that the area is 128 3 , we have 8 3 k^2 = 128 3 k^2 = 16 Since k > 0 , we get k = 4 . Substituting k = 4 : ⁻¹ ( 2 k ) = ⁻¹ ( 2 4 ) = ⁻¹ ( 1 2 ) = 6 Answer: 6