• # question_answer For the curve, $x={{t}^{2}}-1$, $y={{t}^{2}}-t,$ the tangent line is perpendicular to x-axis, when A)  $t=0$            B)  $t=\infty$ C)  $t=\frac{1}{\sqrt{3}}$                     D)  $t=\frac{-1}{\sqrt{3}}$

Given, $x={{t}^{2}}-1$ and $y={{t}^{2}}-t$ Now, $\frac{dx}{dt}=2t$ and $\frac{dy}{dt}=2t-1$ $\therefore$    $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{2t-1}{2t}$ $\because$ The tangent line is perpendicular to x-axis. $\therefore$    $\frac{dy}{dx}=\tan \,{{90}^{o}}=\infty =\frac{1}{0}$ $\Rightarrow$            $\frac{2t-1}{2t}=\frac{1}{0}$ $\Rightarrow$            $t=0$ where, $\left( t\ne \frac{1}{2} \right)$