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 ........

Prove that the curves $x=y^{2}$ and $xy=k$ cut at right angles if $8k^{2}=1$.

Let $m$ be the slope of the normal $L$ drawn at $(1, 2)$ to the curve $x = t^2 - 7t + 7, y = t^2 - 4t - 10$ and $ax + by + c = 0$ be the equation of the normal $L$. If the $G$.$C$.$D$. of $(a, b, c)$ is $1$,then $m(a + b + c) =$

The tangent to the curve $y = e^{2x}$ at the point $(0, 1)$ meets the $x$-axis at

The $x-$intercept of the tangent at any arbitrary point of the curve $\frac{a}{x^2} + \frac{b}{y^2} = 1$ is proportional to

Let the quadratic curve passing through the point $(-1, 0)$ and touching the line $y = x$ at $(1, 1)$ be $y = f(x)$. Then the $x$-intercept of the normal to the curve at the point $(\alpha, \alpha + 1)$ in the first quadrant is $..........$.