Find the point at which the tangent to the curve $y = \sqrt{4x - 3} - 1$ has its slope $\frac{2}{3}$.

If $\theta$ denotes the acute angle between the curves $y = 10 - x^2$ and $y = 2 + x^2$ at a point of their intersection,then $|\tan \theta |$ is equal to

$A$ curve is represented by the equations $x = \sec^2 t$ and $y = \cot t$,where $t$ is a parameter. If the tangent at the point $P$ on the curve where $t = \pi/4$ meets the curve again at the point $Q$,then the $x$-coordinate of $Q$ is equal to

The length of the subtangent at any point $(x_1, y_1)$ on the curve $y=5^x$ is

At which point does the tangent to the curve $\sqrt{x} + \sqrt{y} = \sqrt{a}$ become perpendicular to the $x$-axis?

Let $\ell$ be a line which is normal to the curve $y=2x^2+x+2$ at a point $P$ on the curve. If the point $Q(6,4)$ lies on the line $\ell$ and $O$ is the origin,then the area of the triangle $OPQ$ is equal to.......