The length of the normal to the curve $x = a(\theta + \sin \theta)$,$y = a(1 - \cos \theta)$ at the point $\theta = \pi/2$ is

What is the angle between the curves $y = x^2$ and $6y = 7 - x^3$ at the point $(1, 1)$?

The equation of the tangent to the curve $y = x + \frac{4}{x^2}$ which is parallel to the $X$-axis is:

If the tangent of the curve $y=e^{x}$ at a point $(c, e^{c})$ and the normal to the parabola $y^{2}=4x$ at the point $(1,2)$ intersect at the same point on the $x$-axis,then the value of $c$ is

Find the points on the curve $\frac{x^2}{4} + \frac{y^2}{25} = 1$ at which the tangents are parallel to the $x$-axis.

Show that the equation of the normal at any point on the curve $x=3 \cos \theta-\cos ^{3} \theta, y=3 \sin \theta-\sin ^{3} \theta$ is $4(y \cos ^{3} \theta-x \sin ^{3} \theta)=3 \sin 4 \theta$.