Q39 / 75 · VITEEE 2021 · Mathematics

The angle of intersection to the curve $y=x^2$, $6 y=7-x^3$ at $(1,1)$ is :

  • Correct answer
    A
    $\frac{\pi}{2}$
  • B
    $\frac{\pi}{4}$
  • C
    $\frac{\pi}{3}$
  • D
    $\pi$

Solution

Let $m_1$ and $m_2$ be slope of curve $y=x^2$ and $6 y=7-x^3$ respectively. Now, $y=x^2 \Rightarrow \frac{d y}{d x}=2 x$ $$ \Rightarrow\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{(1,1)}=2 \text { i.e. } \mathrm{m}_1=2 $$ and $6 y=7-x^3 \Rightarrow 6 \frac{d y}{d x}=-3 x^2$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{3}{6} \mathrm{x}^2=-\frac{1}{2} \mathrm{x}^2$ $\Rightarrow\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{(1,1)}=-\frac{1}{2}(1)^2=-\frac{1}{2}$ $\therefore \mathrm{m}_2=-\frac{1}{2} \therefore \mathrm{m}_1 \mathrm{~m}_2=2 .-\frac{1}{2}=-1$ $\therefore \quad$ Angle of intersection is $90^{\circ}$ i.e. $\frac{\pi}{2}$

Hence, the correct answer is option: 'A'.

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