Question

If the tangent to the curve $x={\mathrm{at}}^2,y=2\mathrm{at}$ is perpendicular to x-axis, then its point of contact is .

• A. (a, a)
• B. (0, a)
• C. (0, 0)
• D. (a, 0)

Answer 1

The correct option is C (0, 0)

Let P(h, k) be the point where the tangent to the curve $x={\mathrm{at}}^2,y=2\mathrm{at}$ is perpendicular to x-axis.
$\therefore {\left(\frac{\mathrm{dx}}{\mathrm{dy}}\right)}_P=0\mathrm{Given}\mathrm{that}x={\mathrm{at}}^2\mathrm{and}y=2\mathrm{at}\Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}}=2\mathrm{at}\mathrm{and}\frac{\mathrm{dy}}{\mathrm{dt}}=2a\Rightarrow \frac{\mathrm{dx}}{\mathrm{dy}}=\frac{\frac{\mathrm{dx}}{\mathrm{dt}}}{\frac{\mathrm{dy}}{\mathrm{dt}}}=t\because {\left(\frac{\mathrm{dx}}{\mathrm{dy}}\right)}_P=0\Rightarrow t=0\therefore \mathrm{Co}-\mathrm{ordinates}\mathrm{of}\mathrm{the}\mathrm{required}\mathrm{point}\mathrm{is}\left(0,0\right).$