The tangent to the curve $y = ax^2 + bx$ at $(2, -8)$ is parallel to the $x$-axis. Then:

If the lengths of the tangent,subtangent,normal and subnormal for the curve $y=x^2+x-1$ at the point $(1,1)$ are $a, b, c$ and $d$ respectively,then their increasing order is

If the angle made by the tangent at the point $(x_{0}, y_{0})$ on the curve $x=12(t+\sin t \cos t)$,$y =12(1+\sin t )^{2}$,$0 < t < \frac{\pi}{2}$,with the positive $x$-axis is $\frac{\pi}{3}$,then $y_{0}$ is equal to

If an equation of a tangent to the curve $y = \cos(x + y)$,where $-1 - \pi \le x \le 1 + \pi$,is $x + 2y = k$,then $k$ is equal to

If a normal drawn at a point $P$ to the curve $y=\sin x$ passes through the origin,then the locus of $P$ is

$x_1, x_2 \in N$. 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 $Q(x_2, y_2)$, then $x_1x_2+y_1y_2=$