The angle of intersection of the parabolas

Question:

The angle of intersection of the parabolas $y^{2}=4$ ax and $x^{2}=4$ ay at the origin is

A. $\frac{\pi}{6}$

B. $\frac{\pi}{3}$

C. $\frac{\pi}{2}$

D. $\frac{\pi}{4}$

Solution:

Given that the the parabolas $y^{2}=4 a x$ and $x^{2}=4 a y$

Differentiating both w.r.t. $\mathrm{x}$,

$2 y \frac{d y}{d x}=4 a$ and $2 x=4 a \frac{d y}{d x}$

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

At origin,

$m_{1}=$ infinity and $m_{2}=0$

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

$\Rightarrow \tan \theta=\left|\frac{\infty-0}{1+0 \times \infty}\right|=\infty$

$\Rightarrow \theta=90^{\circ}$

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