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

The correct option is C $\frac{\pi }{2}$

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 with respect to $x$, 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 $(0,0)$

$m_1=\frac{dy}{dx}=\infty$ and $m_2=\frac{dy}{dx}=0$

$\therefore \tan \theta =\frac{m_2-m_1}{1+m_1{m}_2}=\infty$

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