Question:
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. $45^{\circ}$
Solution:
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
$\tan \theta=\left|\frac{\mathrm{m}_{1}-\mathrm{m}_{2}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}\right|$
$\Rightarrow \tan \theta=\left|\frac{\frac{1}{2}-2}{1+1}\right|$
$\Rightarrow \tan \theta=\frac{3}{4}$
$\Rightarrow \theta=\tan ^{-1} \frac{3}{4}$