Question

For the curve $x=t^2-1$, $y=t^2-t$, the tangent is perpendicular to $x$-axis then

• A. $t=0$
• B. $t=\frac{1}{2}$
• C. $t=1$
• D. $t=\frac{1}{\sqrt{3}}$

Answer 1

The correct option is A $t=0$

Given curve is $x=t^2-1,y=t^2-t$

Derivating w.r.to $t$ we get

$\frac{dx}{dt}=2t$ -------(1)

and $\frac{dy}{dt}=2t-1$ -------(2)

dividing (2) by (1) we get

$\frac{dy}{dx}=\frac{2t-1}{2t}$

Therefore, the slope of the tangent is $\frac{2t-1}{2t}$

Given that the tangent is perpendicular to x-axis. Therefore, tangent is parallel to y-axis.

We know that slope of y-axis is infinity and the slopes of the two parallel lines are equal.

Therefore, slope of the tangent is infinity.

Hence, $\frac{2t-1}{2t}=\frac{1}{0}$

$⟹2t=0⟹t=0$