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

If the tangent drawn to the curve $y=x^3$ at a point $(\alpha, \beta)$ cuts the curve again at another point $(\alpha_1, \beta_1)$,then $\frac{\beta_1}{\beta}=$

If a line having slope $2$ is a tangent to the curve $y=x^4-6x^3+13x^2-10x+5$ at points $P(x_1, y_1)$ and $Q(x_2, y_2)$, where $x_1, x_2 \in \mathbb{N}$, then $x_1x_2 - y_1y_2 =$

The values of parameter $a$ such that the line $(\log_{2}(1 + 5a - a^{2}))x - 5y - (a^{2} - 5) = 0$ is a normal to the curve $xy = 1$ may lie in the interval

$A$ curve is represented by the equations $x = \sec^2 t$ and $y = \cot t$,where $t$ is a parameter. If the tangent at the point $P$ on the curve where $t = \pi/4$ meets the curve again at the point $Q$,then the $x$-coordinate of $Q$ is equal to

For any point on a curve,what is $\sqrt{\frac{\text{subnormal}}{\text{subtangent}}}$ equal to?