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 $x_1x_2 - y_1y_2 =$

If $\theta$ is an angle between the curves $x^2+4y=0$ and $xy=2$,then $\tan \theta=$

The angle between the tangents drawn at $(0,0)$ to the curves $y^3-x^2 y+5 y-2 x=0$ and $x^4-x^3 y^2+5 x+2 y=0$ is

If $P$ and $Q$ are two different points on the curve $y = x^3 - x$ such that the tangent at $P$ intersects the curve again at $Q$,then $\frac{m_{OQ} + 1}{m_{OP} + 1}$ is equal to,where $O$ is the origin and $m_{AB}$ represents the slope of the line segment $AB$.

The equation of the tangent to the curve $y=\pi e^{\frac{-x}{\pi}}$ at the point where it crosses the $y$-axis is

If the length of the subnormal at any point of the curve $y^n = a^{n-1}x$ is constant,then $n$ equals-