Find the length of the subnormal of the parabola $y^2 = 16x$ at the point where the $x$-coordinate is $4$.

The area of the triangle formed by the lines joining the vertex of the parabola $x^2 = 20y$ to the ends of its latus rectum is

If $a > 0$ and $b^2 - 4ac = 0$,then the curve $y = ax^2 + bx + c$

Let the curve $C$ be the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y=-5$,then the distance between $A$ and $B$ is

Find the equation of the parabola that satisfies the following conditions: Vertex $(0, 0)$,passing through $(2, 3)$,and the axis is along the $x$-axis.

Statement $-1:$ The line $x - 2y = 2$ meets the parabola $y^2 + 2x = 0$ only at the point $(-2, -2).$
Statement $-2:$ The line $y = mx - \frac{1}{2m}$ $(m \neq 0)$ is tangent to the parabola $y^2 = -2x$ at the point $\left( -\frac{1}{2m^2}, -\frac{1}{m} \right).$