Tangents and it...

If the curves $$\dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{4} = 1$$ and $$y^{3} = 16x$$ intersect at right angles, then $$a^{2}$$ is equal to

If the slope of one of the lines given by $${a^2}{x^2} + 2hxy+by^2 = 0$$ be three times of the other , then h is equal to 

The equation of normal to the curve $$y=log^x_e$$ at the point $$P(1, 0)$$ is ___________.

If the line joining the point (0, 3 ) and (5, -2) is a tangent to the curve $$y= \dfrac{c}{x+1}$$, then the value of c is 

If the tangent at any point on the curve $$x+y^4=a$$ cuts off intercepts p and q on the co-ordinate axes, the value of $$p^{-4/3}+q^{-4/3}$$ is 

Let N be the set of positive integers. For all $$n \in N$$, let
$$f_n = (n + 1)^{1/3} - n^{1/3}$$ and $$A = \left\{n \in N : f_{n +1} < \dfrac{1}{3(n + 1)^{2/3}} < f_n \right\}$$
Then

At any point on the curve $$2x^{2}y^{2}-x^{4}=c$$, the mean proportional between the abscissa and the difference between the abscissa and the sub-normal drawn to the curve at the same point is equal to 

Given $$g(x)= \dfrac{x+2}{x-1}$$ and the line 3x + y -10 =0, then the line is 

The angle formed bt the positive y-axis and the tangent to $$y = x^{2}+4x-17$$ at $$(5/2, -3/4)$$ is