Mathematics - Application of Derivative Question with Solution | TestHub

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