If the normal to the curve $y = f(x)$ is parallel to the $x-$axis,then the correct statement is:

The equation of the normal to the curve $y^3 + 2xy + x^3 = (x - 1)^3$ at the point $(1, -1)$ 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 $AB$.

If $ST$ and $SN$ are the lengths of the subtangent and the subnormal at the point $\theta = \frac{\pi}{2}$ on the curve $x = a(\theta + \sin \theta ), y = a(1 - \cos \theta ), a \neq 0$,then:

$A(1, -3)$ and $B(4, 3)$ are two points on the curve $y = x - \frac{4}{x}$. The points on the curve,the tangents at which are parallel to the chord $AB$,are

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