Let $A$ be the vertex and $L$ be the length of the latus rectum of the parabola $y^2 - 2y - 4x - 7 = 0$. Find the equation of the parabola with $A$ as the vertex,$2L$ as the length of the latus rectum,and the axis at right angles to that of the given curve.

At which point does the line $x + y = 1$ intersect the parabola $y = x - x^2$?

$A$ line passing through the point of intersection of $x+y=4$ and $x-y=2$ makes an angle $\tan^{-1}\left(\frac{3}{4}\right)$ with the $X$-axis. It intersects the parabola $y^{2}=4(x-3)$ at points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$, respectively. Then $|x_{1}-x_{2}|$ is equal to

Let $A_1, B_1, C_1$ be three points in the $xy$-plane. Suppose that the lines $A_1 C_1$ and $B_1 C_1$ are tangents to the curve $y^2=8x$ at $A_1$ and $B_1$,respectively. If $O=(0,0)$ and $C_1=(-4,0)$,then which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ The length of the line segment $OA_1$ is $4\sqrt{3}$
$(B)$ The length of the line segment $A_1 B_1$ is $16$
$(C)$ The orthocenter of the triangle $A_1 B_1 C_1$ is $(0,0)$
$(D)$ The orthocenter of the triangle $A_1 B_1 C_1$ is $(1,0)$

What is the equation of the directrix of the parabola $y^2 + 4y + 4x + 2 = 0$?

The focus of a parabolic mirror as shown in the figure is at a distance of $5 \, cm$ from its vertex. If the mirror is $45 \, cm$ deep,find the distance $AB$. (in $, cm$)