The $x$-coordinate of the point on the curve $y = a\left( {{e^{\frac{x}{a}}} + {e^{ - \frac{x}{a}}}} \right)$ where the tangent is parallel to the $x$-axis is ........

Let $a$ be a fixed positive real number and $n$ be an arbitrary constant. For the curve $y = \frac{x^n}{a^{n-1}}$, if the length of the subnormal at any point $(\alpha, \beta)$ is proportional to $a^2$, then $n =$

Find the $x$-coordinate of the point on the curve $ay^2 = x^3$ where the normal makes equal intercepts on the axes.

The number of tangents to the curve $y^2(x-a)=x^2(x+a)$ $(a>0)$ that are parallel to the $X$-axis is

The area of the triangle formed by the coordinate axes and the normal to the curve $y = e^{2x} + x^2$ at the point $x = 0$.

The slope of the normal drawn at a point $P$ to the curve $y = x^3 - 10x^2 + 31x - 30$ is $-\frac{1}{14}$. Find the $x$-intercept of the tangent at point $P$, given that the $x$-coordinate of $P$ is an integer.