The angle between the curves


The angle between the curves $y^{2}=x$ and $x^{2}=y$ at $(1,1)$ is

A. $\tan ^{-1} \frac{4}{3}$

B. $\tan ^{-1} \frac{3}{4}$

C. $90^{\circ}$

D. $45^{\circ}$


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

Differentiating both the equations w.r.t. $x$,

$\Rightarrow 2 y \frac{d y}{d x}=1$ and $2 x=\frac{d y}{d x}$

$\Rightarrow \frac{d y}{d x}=\frac{1}{2 y}$ and $\frac{d y}{d x}=2 x$

For $(1,1)$ :

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}$ and $\frac{\mathrm{dy}}{\mathrm{dx}}=2$

Thus we get

$\tan \theta=\left|\frac{\mathrm{m}_{1}-\mathrm{m}_{2}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}\right|$

$\Rightarrow \tan \theta=\left|\frac{\frac{1}{2}-2}{1+1}\right|$

$\Rightarrow \tan \theta=\frac{3}{4}$

$\Rightarrow \theta=\tan ^{-1} \frac{3}{4}$

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