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

Find the equation of the tangent to the curve $y = \frac{x-7}{(x-2)(x-3)}$ at the point where it cuts the $x$-axis.

If the tangent to the curve $xy+ax+by=0$ at $(1,1)$ makes an angle $\tan^{-1} 2$ with the $x$-axis, then $\frac{a+b}{ab} =$

The angle between the curve $2y = e^{-x/2}$ and the $y$-axis is $\tan^{-1}(k)$,then $k = $

If curves $\frac{x^2}{a^2} + \frac{y^2}{4} = 1$ and $y^3 = 16x$ intersect orthogonally,then $a^2$ equals:

Let $C$ be a curve given by $y(x)=1+\sqrt{4x-3}$,$x>\frac{3}{4}$. If $P$ is a point on $C$ such that the tangent at $P$ has a slope of $\frac{2}{3}$,then a point through which the normal at $P$ passes is: