If the equation of the tangent at $(2,3)$ on the curve $y^2 = ax^3 + b$ is $y = 4x - 5$,then the value of $a^2 + b^2$ is:

The normal to the curve $x=a(1+\cos \theta)$,$y=a \sin \theta$ at $\theta$ always passes through a fixed point. Find the fixed point.

The area of the triangle formed by any tangent to the curve $xy = 100$ with the coordinate axes is:

The equation of a straight line passing through the point $(3, 6)$ and cutting the curve $y = \sqrt{x}$ orthogonally is

The lengths of tangent, subtangent, normal and subnormal for the curve $y=x^2+x-1$ at $(1,1)$ are $A, B, C$ and $D$ respectively, then their increasing order is

If two curves $x=y^2$ and $xy=a^3$ cut each other orthogonally at a point,then $a^2$ is equal to