For the curve $b{y^2} = {(x + a)^3}$,the square of the subtangent is proportional to:

If at any point $(x_1, y_1)$ on the curve $y=f(x)$ the lengths of the subtangent and subnormal are equal, then the length of the tangent drawn to that curve at that point is

Find the point on the curve $y=x^{3}-11x+5$ at which the tangent is $y=x-11$.

The slope of the normal drawn at a point $P$ to the curve $y = x^3 - 10x^2 + 31x - 30$ is $-\frac{1}{14}$. Find the $x$-intercept of the tangent at point $P$, given that the $x$-coordinate of $P$ is an integer.

If the normal drawn at a point $P$ on the curve $3y = 6x - 5x^3$ passes through $(0,0)$,then the positive integral value of the abscissa of the point $P$ 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