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