For the curve ${y^n} = {a^{n - 1}}x$,the subnormal at any point is constant. The value of $n$ must be

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:

If the normal to the curve $y = x + \frac{2}{x}$ at the point where the abscissa is $2$ meets the coordinate axes at points $A$ and $B$,find the length of $AB$.

If the normal drawn at the point $P$ on the curve $y^2 = x^3 - x + 1$ makes equal intercepts on the coordinate axes,then the equation of the tangent drawn to the curve at $P$ is

The equation of the tangent to the curve $(1+x^2)y = 2-x$,where it crosses the $X$-axis,is

Let $a, b, c \in \mathbb{R}$ be such that $2a + 3b + 6c = 0$ and $g(x)$ be the antiderivative of $f(x) = ax^2 + bx + c$. If the slopes of the tangents drawn to the curve $y = g(x)$ at $(1, g(1))$ and $(2, g(2))$ are equal,then