Question

The angle of intersection of the parabolas y² = 4 ax and x² = 4ay at the origin is

(a) π/6
(b) π/3
(c) π/2
(d) π/4

Answer 1

(c) π/2

$$\begin{gathered} \mathrm{Given}: \\ {y}^2=4ax...\left(1\right) \\ {x}^2=4ay...\left(2\right) \\ \text{Point =}\left(0,0\right) \\ \mathrm{On}\mathrm{d}\text{ifferentiating (1) w.r.t.}\mathit{\text{x}}\text{,}\text{we get} \\ 2y\frac{dy}{dx}=4a \\ \Rightarrow \frac{dy}{dx}=\frac{2a}{y} \\ \Rightarrow m_1=\infty \\ \mathrm{Now},\mathrm{on}\mathrm{d}\text{ifferentiating (2) w.r.t.}\mathit{\text{x}}\text{, we get} \\ \mathit{2}x=4a\frac{dy}{dx} \\ \Rightarrow \frac{dy}{dx}=\frac{x}{2a}=0 \\ \therefore \tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|=\left|\frac{\infty }{1+0}\right|=\infty \\ \Rightarrow \theta ={\tan }^{-1}\infty =\frac{\mathrm{\pi }}{2} \end{gathered}$$