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?