Tangents and it...

At origin the curve  $$y^{2}=x^{3}+x$$

Observe the following statements
I: If $$p$$ and $$q$$ are the lengths of perpendiculars from the origin on the tangent and normal at any point on the curve $$x^{\frac{2}{3}}+y^{\frac{2}{3}}=1$$ then $$4p^{2}+q^{2}=1$$.
II: If the tangent at any point $$P$$ on the curve $$x^{3}.y^{2}=a^{5}$$ cuts the coordinate axes at $$A$$ and $$B$$ then $$AP : PB = 3 : 2$$

Observe the following statements for the curve $$x
= at^{3}$$, $$y = at^{4}$$ at $$t = 1$$.
I : The equation of the tangent to the curve is $$4x-3y- a = 0$$
II : The equation of the normal to the curve is $$3x +4y-  7a = 0$$
III: Angle between tangent and normal at any point on the curve is $$\displaystyle \frac{\pi}{2}$$
Which of the above statements are correct.

Assertion (A): The normal to the curve $$ay^{2}=x^{3}(a\neq 0, x\neq 0)$$ at a point $$(x, y)$$ on it makes equal intercepts on the axes, then $$\displaystyle x=\frac{4a}{9}$$.
Reason (R): The normal at $$(x_{1}, y_{1})$$ on the curve $$y=f(x)$$ makes equal intercepts on the coordinate axes, then $$\left.\displaystyle \frac{dy} {dx}\right|_{(x_{1},y_{1})}=1$$

Assertion(A): If the tangent at any point $$P$$ on the curve $$xy = a^{2}$$ meets the axes at $$A$$ and $$B$$ then $$AP : PB = 1 : 1$$
Reason(R): The tangent at $$P(x, y)$$ on the curve $$X^{m}.Y^{n}=a^{m+n}$$ meets the axes at $$A$$ and $$B$$. Then the ratio of $$P$$ divides $$\overline{AB}$$ is $$n : m$$.

The point of intersection of the tangents drawn to the curve $$x^{2}y=1-y$$ at the points where it is met by the curve $$xy=1-y$$ is given by

The number of tangents to the curve $$x^{3/2}+y^{3/2}=a^{3/2}$$, where the tangents are equally inclined to the axes, is

The points on the hyperbola $$x^{2}-y^{2}=2$$ closest to the point (0, 1) are

If the circle $$x^2 + y^2 + 2gx + 2fy + c =0$$ is touched by y = x at P in the first quadrant, such that $$OP = 6 \sqrt2$$, then the value of $$c$$ is

lf the curve $$y=px^{2}+qx+r$$ passes through the point (1, 2) and the line $$y=x$$ touches it at the origin, then the values of $$p,\ q$$ and $$r$$ are