The slope of the tangent of the curve $\left(\frac{x}{31}\right)^n + \left(\frac{y}{1209}\right)^n = 2$ at $(31, 1209)$ 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 of the curve $4y^3 = 3ax^2 + x^3$ drawn at the point $(a, a)$ forms a triangle of area $\frac{25}{24}$ sq. units with the coordinate axes,then $a =$

The angle between the curves $y^2=x$ and $x^2=y$ at the point $(1,1)$ is

If the tangent to the curve $y = \frac{x}{x^2-3}$,$x \in R, (x \neq \pm \sqrt{3})$ at a point $(\alpha, \beta) \neq (0,0)$ on it,is parallel to the line $2x + 6y - 11 = 0$,then

The length of the tangent drawn at the point $P\left(\frac{\pi}{4}\right)$ on the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2^{\frac{2}{3}}$ is