Tangents and it...

If the sum of the squares of the intercepts on the axes cut off by the tangent to the cuve $\displaystyle x^{1/3}+y^{1/3}= a^{1/3}\left ( a> 0 \right )$ at $\displaystyle \left ( \dfrac{a}{8}, \dfrac{a}{8} \right )$ is $2$, then $a$ has the value

The coordinates of the point $P$ on the curve $y^{2}= 2x^{3}$ the tangent at which is perpendicular to the line $4x-3y + 2 = 0$, are given by

The tangent to the curve $x= a\sqrt{\cos 2\theta }\cos \theta$, $$y= a\sqrt{\cos 2\theta }\sin \theta
$at the point corresponding to$\theta = \pi /6$$ is

The equation of the tangent line at an inflection point of $\displaystyle f\left ( x \right )=x^{4}-6x^{3}+12x^{2}-8x+3$ is

The equation of the common tangent to the curves $\displaystyle y^{2}= 8x$ and $\displaystyle xy= -1$ is

The point(s) on the curve $y^{3}+3x^{2}=12y$ the tangent is vertical is (are)

The equation of the tangents to $\displaystyle 4x^{2}-9y^{2}=36$ which are perpendicular to the straight line $\displaystyle 2y+5x= 10$ are

Of all the line tangent to the graph of the curve  $\displaystyle y=\frac{6}{x^{2}+3},$ find the equations of the tangent lines of minimum and maximum slope respectively.

If the tangent is drawn to the curve y = f(x) at a point where it crosses the y - axis then its equation is

What normal to the curve $\displaystyle y=x^{2}$ form the shortest chord?