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

The sum of the intercepts on the axes of the tangent to the curve $\sqrt{x} + \sqrt{y} = \sqrt{a}$ is equal to:

The normal to the curve $y(x-2)(x-3)=x+6$ at the point,where the curve intersects the $Y$-axis,passes through the point

If the curves $y = x^3 - 3x^2 - 8x - 4$ and $y = 3x^2 + 7x + 4$ touch each other at a point $P$,then the equation of the common tangent at $P$ is:

Find the condition for the curves $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ and $xy=c^{2}$ to intersect orthogonally.

$P$ is the point of contact of the tangent drawn from the origin to the curve $y = \log_{e} x$. The length of the perpendicular drawn from the origin to the normal at $P$ is