The line joining the points $(0,3)$ and $(5,-2)$ is a tangent to the curve $y=\frac{c}{x+1}$. Then,$c=$

On the curve $y=x^3$,the point at which the tangent line is parallel to the chord joining the points $(-1, -1)$ and $(2, 8)$ is

If the angle between the curves $y^2=4x$ and $y=e^{-x/2}$ is $\theta$,then $\operatorname{cosec}^2(\theta/2)=$

If the tangent of the curve $y=e^{x}$ at a point $(c, e^{c})$ and the normal to the parabola $y^{2}=4x$ at the point $(1,2)$ intersect at the same point on the $x$-axis,then the value of $c$ is

The equation of the tangent to the curve $y = \sqrt{2} \sin \left(2x + \frac{\pi}{4}\right)$ at $x = \frac{\pi}{4}$ is:

Let the curve be represented by $x=2(\cos t+t \sin t)$ and $y=2(\sin t-t \cos t)$. Then the normal at any point '$t$' of the curve is at a distance of . . . . . . units from the origin.