If the normal drawn at the point $P$ on the curve $y=x \log x$ is parallel to the line $2x-2y=3$,then $P=$

If the tangent to the curve $y=x^{3}$ at the point $P(t, t^{3})$ meets the curve again at $Q$,then the ordinate of the point which divides $PQ$ internally in the ratio $1:2$ is

The curve given by $x + y = e^{xy}$ has a tangent parallel to the $y-$ axis at the point

What is the equation of the tangent to the parabola $y = 2 + 4x - 4x^2$ having a slope of $-4$?

Show that the normal at any point $\theta$ to the curve $x=a \cos \theta+a \theta \sin \theta, y=a \sin \theta-a \theta \cos \theta$ is at a constant distance from the origin.

$y=f(x)$ and $x=g(y)$ are two curves and $P(x, y)$ is a common point of the two curves. If at $P$, on the curve $y=f(x)$, $\frac{dy}{dx}=Q(x)$ and at the same point $P$ on the curve $x=g(y)$, $\frac{dx}{dy}=-Q(x)$, then