The angle between the curves $y^2=4x+4$ and $y^2=36(9-x)$ is (in $^{\circ}$)

If the curve $y = ax^{2} + bx + c, x \in R,$ passes through the point $(1, 2)$ and the tangent line to this curve at the origin is $y = x$,then the possible values of $a, b, c$ are:

If $f: R \rightarrow R$ is a function defined for all $x \in R$ by $f(x)=x^3+f^{\prime}(1) x^2+f^{\prime \prime}(2) x-f^{\prime \prime \prime}(3)$,then the area (in sq. units) of the triangle formed by the $X$-axis,the tangent,and the normal drawn to the curve $y=f(x)$ at $x=0$ is

Find the points on the curve $9y^2 = x^3$ where the normal to the curve makes equal intercepts with the axes.

If the curves $y^2 = 12x - 3$ and $y^2 = 12 - kx$ cut each other orthogonally, then the length of the sub-tangent at $(1, b)$ on the curve $y^2 = 12 - kx$ is

If $ST$ and $SN$ are the lengths of the subtangent and the subnormal at the point $\theta = \frac{\pi}{2}$ on the curve $x = a(\theta + \sin \theta ), y = a(1 - \cos \theta ), a \neq 0$,then: