Question

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

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

Answer 1

The correct option is C $t=\frac{1}{\sqrt{3}}$

$\because x=acos\theta \Rightarrow \frac{dx}{d\theta }=-asin\theta andy=bsin\theta \Rightarrow \frac{dy}{d\theta }=bcos\theta \therefore \frac{dy}{dx}=-\frac{b}{a}cot\theta \Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{b}{a}cose{c}^2\theta \frac{d\theta }{dx}=-\frac{b}{a^2}cose{c}^3\theta \therefore \frac{d^{3}y}{d{x}^3}=\frac{3b}{a^2}cose{c}^2\theta (-cosec\theta cot\theta )\frac{d\theta }{dx}=\frac{3b}{a^2}cose{c}^3\theta cot\theta (-\frac{1}{asin\theta })=-\frac{3b}{a^3}cose{c}^4\theta cot\theta$