The area of the region enclosed between parabola $y^{2}=x$ and the line $y=m x$ is $\frac{1}{48}$ Then, the value of $m$ is

• Correct answer
  A

  -2
• B

  -1
• C

  1
• Correct answer
  D

  2

Solution

Equation of parabola is $y^{2}=x$ and line $y=m x$ For intersection point of both curves put $x=y^{2}$, we get $$ y=m y^{2} \Rightarrow y(m y-1)=0 $$ $\Rightarrow \quad y=0$ or $y=\frac{1}{m}$ Then, $\quad x=0$ or $x=\frac{1}{m^{2}}$ $\therefore$ Intersection points are (0,0) and $P\left(\frac{1}{m^{2}}, \frac{1}{m}\right)$ $\therefore$ Required area $=\int_{0}^{1 / m}\left(\frac{y}{m}-y^{2}\right)|d y=|\left[\frac{y^{2}}{2 m}-\frac{y^{3}}{3}\right]_{0}^{1 / m} \mid$ $=\left|\frac{1}{2 m^{3}}-\frac{1}{3 m^{3}}\right|=\left|\frac{1}{6 m^{3}}\right|=\frac{1}{48}$ $\Rightarrow \quad \frac{1}{6 m^{3}}=\pm \frac{1}{48} \Rightarrow m^{3}=\pm 8$ (given) Now, if $m^{3}=8$ $\Rightarrow \quad m^{3}=(2)^{3} \Rightarrow m=2$ If $\quad m^{3}=-8$ $\Rightarrow \quad m^{3}=(-2)^{3} \Rightarrow m=-2$

Hence, the correct answer is options: 'A', 'D'.