If the tangent to the curve $2y^3 = ax^2 + x^3$ at the point $(a, a)$ cuts the coordinate axes at $p$ and $q$ such that $p^2 + q^2 = 61$,then what is the value of $a$?

If the straight line $x \cos \alpha + y \sin \alpha = p$ touches the curve $(\frac{x}{a})^n + (\frac{y}{b})^n = 2$ at the point $(a, b)$ on it and $\frac{1}{a^2} + \frac{1}{b^2} = \frac{k}{p^2}$,then $k =$

If the tangent at the point $P$ with coordinates $(h, k)$ on the curve $y^{2}=2x^{3}$ is perpendicular to the straight line $4x=3y$, then

If the normal to the curve $y = x + \frac{2}{x}$ at the point where the abscissa is $2$ meets the coordinate axes at points $A$ and $B$,find the length of $AB$.

Let $P(h, k)$ be a point on the curve $y=x^{2}+7x+2$ nearest to the line $y=3x-3$. Then the equation of the normal to the curve at $P$ is:

The value of $c$ such that the straight line joining the points $(0,3)$ and $(5,-2)$ is tangent to the curve $y=\frac{c}{x+1}$ is