Mathematics - Application of Derivative Question with Solution | TestHub

Let $x=64{\cos }^{2}t$.

$$\begin{array}{rl}y& =54{\sin }^t\\ \frac{dx}{dt}& =-192\cos t(-\sin t)\end{array}$$

Equation of tangent at $t$:

$$\begin{array}{rl}\Rightarrow y-64{\sin }^{2}t& =\frac{-\sin t}{\cos t}(x-64{\cos }^{2}t)\\ \Rightarrow \frac{y}{\sin t}-64\sin t& =\frac{-\sin t}{\cos t}(x-64{\cos }^{2}t)\\ \Rightarrow \frac{x}{64\cos t}+\frac{y}{64\sin t}& =1\\ \Rightarrow Segmentoftangentbetween\alpha and\beta & =\sqrt{(64{)}^2{\cos }^{2}t+64^2{\sin }^{2}t}\\ & =64\\ Length& =64=x^2\therefore \lambda =8\end{array}$$