Question

Write the angle between the curves y² = 4x and x² = 2y − 3 at the point (1, 2).

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ {y}^2=4x...\left(1\right) \\ {x}^2=2y-3...\left(2\right) \\ \mathrm{On}\mathrm{d}\text{ifferentiating (1) w.r.t.}\mathit{\text{x}}\text{, we get} \\ \mathit{2}y\frac{dy}{dx}=4 \\ \Rightarrow \frac{dy}{dx}\mathit{=}\frac{2}{y} \\ \mathit{\Rightarrow }{m}_{\mathit{1}}={\left(\frac{dy}{dx}\right)}_{\left(1,2\right)}=\frac{2}{2}=1 \\ \mathrm{On}\mathrm{d}\text{ifferentiating (2) w.r.t.}\mathit{\text{x}}\text{, we get} \\ \mathit{2}x=2\frac{dy}{dx} \\ \mathit{\Rightarrow }\frac{dy}{dx}\mathit{=}x \\ \mathit{\Rightarrow }{m}_{\mathit{2}}={\left(\frac{dy}{dx}\right)}_{\left(1,2\right)}=1 \\ \mathrm{Thus},\mathrm{we}\mathrm{get} \\ \tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right| \\ \Rightarrow \tan \theta =\left|\frac{\mathit{1}\mathit{-}\mathit{1}}{1+1}\right| \\ \Rightarrow \tan \theta =0 \\ \Rightarrow \theta \mathit{=}{0}^{\mathrm{o}} \end{gathered}$$