Tangent and Normal

Water is being poured at the rate of $36 \mathrm{~m}^{3} / \mathrm{min}$ into a cylindrical vessel, whose circular base is of radius 3 meters. Then the water level in the cylinder is rising at the rate of

The abscissa of the point on the curve $y=a\left(e^{\dfrac{x}{a}}+e^{-\dfrac{x}{a}}\right)$ where the tangent is parallel to the X -axis is

If the normal to the curve $y=\mathrm{f}(x)$ at the point $(3,4)$ makes an angle of $\left(\dfrac{3 \pi}{4}\right)$ with the positive $X$-axis, then the value of $f^{\prime}(3)$ is

Find the angle between the tangent to the curve $y^{2}=2ax$ at the points where $x=a/2$.

The equation of the tangent to the curve $y=3 x^{3}-3 x^{2}+x$ at $x=1$ is

The point on the curve $6y=x^{3}+2$ at which $y$-coordinate is changing 8 times as fast as $x$-coordinate is

The equations of the tangent to the curve $x^{2}+y^{2}=10$, where the tangent is parallel to the line $2 x+y-1=0$, are

The point on the curve $4 y^{2}-4 y+2 x-1=0$ at which the tangent becomes parallel to Y axis is

The equation of the tangent to the curve $y=\mathrm{be}^{-x / a}$ at the point where it crosses the Y axis is

The minimum value of the slope of the tangent to curve $y=x^{3}-3 x^{2}+2 x+93$ is

The equation of the tangent to the parabola $y^{2}=8 x$, which is parallel to the line $4 x-y+3=0$ is

The equation of the tangent to the circle, given by $x=5 \cos \theta, y=5 \sin \theta$ at the point $\theta=\dfrac{\pi}{3}$ on it, is

The tangent to the curve $y^{2}-x y+9=0$ is vertical when

If the line $y=4 x-5$ is tangent to the curve $y^{2}=a x^{3}+b$ at the point ( 2,3 ), then the value of $7 a-2 b$ is...

An open tank with a square bottom is to contain 4000 cubic cm . of liquid. The dimensions of the tank so that the surface area of the tank is minimum, is