Question

For any point on the curve $x=4a(t^2-{\cos }^{2}t),y=a(t^2+{\sin }^{2}t),$ which of the following(s) is(are) correct

Answer 1

Gievn curve: $x=4a(t^2-{\cos }^{2}t),y=a(t^2+{\sin }^{2}t)$
$\Rightarrow \frac{dy}{dx}=\frac{dy}{dt}\cdot \frac{dt}{dx}=\frac{a(2t+\sin 2t)}{4a(2t+\sin 2t)}$
$\Rightarrow \frac{dy}{dx}=\frac{1}{4}$
Now,
Length of subtangent at any point $=\frac{y}{\frac{dy}{dx}}=4y$
Length of sub-normal $=y\cdot \frac{dy}{dx}=\frac{y}{4}$