The area of the triangle formed by the normal to the curve $y = e^{2x} + x^2$ at the point $(0, 1)$ with the coordinate axes is $......$ square units.

The line which is parallel to the $x$-axis and intersects the curve $y = \sqrt{x}$ at an angle of $45^\circ$ is given by:

If the tangent to the curve $y = \frac{x}{x^2-3}$,$x \in R, (x \neq \pm \sqrt{3})$ at a point $(\alpha, \beta) \neq (0,0)$ on it,is parallel to the line $2x + 6y - 11 = 0$,then

If $\frac{k}{\alpha^3}$ is the length of the subnormal at any point $P(\alpha, y)$ on the curve $x^2-a^2=\frac{x^2 y^2}{a^2}$,then $k=$

Find the point on the curve $y=x^{3}-11x+5$ at which the tangent is $y=x-11$.

The lines tangent to the curves $y^3 - x^2y + 5y - 2x = 0$ and $x^4 - x^3y^2 + 5x + 2y = 0$ at the origin intersect at an angle $\theta$ equal to