The length of the subtangent to the curve $\sqrt{x} + \sqrt{y} = 3$ at the point $(4, 1)$ is

If the normal to the curve $y(x)=\int_{0}^{x}(2t^{2}-15t+10)dt$ at a point $(a, b)$ is parallel to the line $x+3y=-5$,where $a>1$,then the value of $|a+6b|$ is equal to..........

Find the points on the curve $y=x^{3}$ at which the slope of the tangent is equal to the $y$-coordinate of the point.

$A$ curve is represented by the equations $x = \sec^2 t$ and $y = \cot t$,where $t$ is a parameter. If the tangent at the point $P$ on the curve where $t = \pi/4$ meets the curve again at the point $Q$,then the $x$-coordinate of $Q$ is equal to

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

At which points on the curve $\sqrt{x} + \sqrt{y} = \sqrt{a}$,is the tangent line: $(i)$ parallel to the $x$-axis? $(ii)$ parallel to the $y$-axis? $(iii)$ making equal angles with both axes?