Find the equations of all lines having slope $0$ that are tangent to the curve $y = \frac{1}{x^2 - 2x + 3}$.

$P$ is the point of contact of the tangent drawn from the origin to the curve $y = \log_{e} x$. The length of the perpendicular drawn from the origin to the normal at $P$ is

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 $m(a + b + c) =$

If at any point $(x_1, y_1)$ on the curve $y=f(x)$ the lengths of the subtangent and subnormal are equal, then the length of the tangent drawn to that curve at that point is

The slope of the normal to the curve $y=\frac{x}{x^2+1}$ at $x=-4$ is

The ratio of the length of the subnormal to the square of the length of the subtangent at any point $P$ on the curve $y^2=(2x+1)^3$ is