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 curve $y=x^3-2x^2+3x-4$ intersects the horizontal line $y=-2$ at the point $P(h, k)$. If the tangent drawn to this curve at $P$ meets the $X$-axis at $(x_1, y_1)$, then $x_1=$

$p_1$ and $p_2$ are the perpendicular distances from the origin to the tangent and normal drawn at any point on the curve $x^{2/3} + y^{2/3} = a^{2/3}$ respectively. If $k_1 p_1^2 + k_2 p_2^2 = a^2$,then $k_1 + k_2 =$

The point$(s)$ on the curve $y^3 + 3x^2 = 12y$ where the tangent is vertical (parallel to $y$-axis) is (are):

If the tangent to the curve $y=x+\sin y$ at a point $(a, b)$ is parallel to the line joining $\left(0, \frac{3}{2}\right)$ and $\left(\frac{1}{2}, 2\right),$ then

The length of the subtangent to the curve $x^2 y^2 = a^4$ at the point $(-a, a)$ is