Question

Angle between $y^{2}=x$ and $x^{2}=y$ at the origin is

• A. $2 \, tan^{-1} \left(\frac{3}{4}\right)$
• B. $tan^{-1} \left(\frac{4}{3}\right)$
• C. $\frac{\pi}{2}$
• D. $\frac{\pi}{4}$

Answer 1

Answer: (C)

It is clear from the graph that both the curves have a tangent at the coordinate axes, so the angle between the curves is $\frac{\pi}{2}$



Alternative

Given curves are $y^{2}=x$ and $x^{2}=y$

On differentiating w.r.t. $x$, we get

$2y \frac{dy}{dx}=1$ and $2x=\frac{dy}{dx}$

$\Rightarrow \, \frac{dy}{dx}=\frac{1}{2y}$ and $\frac{dy}{dx}=2x$

At $\left(0, 0\right)$

$m_{1}=\frac{dy}{dx}=\infty$ and $m_{2}=\frac{dy}{dx}=0$

$\therefore \, tan\, \theta =\frac{m_{2}-m_{1}}{1+m_{1}m_{2}}=\infty$

$\Rightarrow \, \theta=\frac{\pi}{2}$