What is the length of the sub-tangent at any point on the curve $y = ax^3$?

If the tangent at a point $P$ on the curve $y=4x^4+x$ is perpendicular to the tangent to the same curve at $(0,0)$,then the point $P$ is

The angle $\theta$,at which the curves $y=3^x$ and $y=7^x$ intersect,is given by

If the point $P(x_1, y_1)$ lying on the curve $y = x^2 - x + 1$ is the closest point to the line $y = x - 3$, then the perpendicular distance from $P$ to the line $3x + 4y - 2 = 0$ is

The sum of the intercepts on the axes of the tangent to the curve $\sqrt{x} + \sqrt{y} = \sqrt{a}$ is equal to:

If $\beta$ is an angle between the normals drawn to the curve $x^2+3y^2=9$ at the points $P(3 \cos \theta, \sqrt{3} \sin \theta)$ and $Q(-3 \sin \theta, \sqrt{3} \cos \theta)$,where $\theta \in (0, \frac{\pi}{2})$,then: