The equation of the tangent to the curve $y=5x^2-3x+7$ at the point $(-1, 4)$ is:

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

Show that the normal at any point $\theta$ to the curve $x=a \cos \theta+a \theta \sin \theta, y=a \sin \theta-a \theta \cos \theta$ is at a constant distance from the origin.

If the tangent to the curve $y = f(x) = x \log_{e} x$ $(x > 0)$ at a point $(c, f(c))$ is parallel to the line segment joining the points $(1, 0)$ and $(e, e)$,then $c$ is equal to:

The normal to the curve $2x^2 + y^2 = 12$ at the point $(2, 2)$ meets the curve again at

The equation of a straight line passing through the point $(3, 6)$ and cutting the curve $y = \sqrt{x}$ orthogonally is