Consider the region $R=\{(x, y) ∈ R^2 : x^2 <y ≤x\}$. If a line $y = α$ divides the area of region R into two equal parts, then which of the following is true?

$3α^2-8α^{3/2}+8=0$

It is given that $y = α$ divides the shaded region into two equal parts.

$∴\int\limits_0^4(x_2-x_1)dy=2\int\limits_0^α(x_2-x_1)dy$

$⇒\int\limits_0^4\left(\sqrt{y}-\frac{y}{2}\right)dy=2\int\limits_0^α\left(\sqrt{y}-\frac{y}{2}\right)dy$

$⇒\left[\frac{2}{3}y^{3/2}-\frac{1}{4}y^2\right]_0^4=2\left[\frac{2}{3}y^{3/2}-\frac{1}{4}y^2\right]_0^α$

$⇒\frac{4}{3}=2\left(\frac{2}{3}α^{3/2}-\frac{1}{4}α^2\right)$

$⇒\frac{2}{3}=\frac{2}{3}α^{3/2}-\frac{1}{4}α^2$

$⇒1=α^{3/2}-\frac{3}{8}α^2$

$⇒8α^{3/2}-3α^2-8=0⇒3α^2-8α^{3/2}+8=0$