The equation of the tangent to the curve $(1 + x^2)y = 2 - x$ at the point where it crosses the $x$-axis is:

The coordinates of the point $P$ on the curve $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$, where the tangent is inclined at an angle $\frac{\pi}{4}$ to the $x$-axis, are

The perpendicular distance from the origin to the normal drawn at any point on the curve $x=a(\cos \theta+\theta \sin \theta), y=a(\sin \theta-\theta \cos \theta)$ is

The coordinates of the point on the curve $9y^2 = x^3$ where the normal to the curve makes equal intercepts with the axes are:

The equation of the tangent to the curve $y=4xe^{x}$ at the point $\left(-1, -\frac{4}{e}\right)$ is:

$A(1, -3)$ 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