Find the equation of the parabola whose focus is $(-1, -2)$ and whose directrix is the line $x - 2y + 3 = 0$.

Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2=x$,such that the area of the region enclosed between the parabola and the chord is $\frac{4}{3}$. Let $R$ denote the region lying in the first quadrant,enclosed by the parabola $y^2=x$,the curve $S$,and the lines $x=1$ and $x=4$. Then which of the following statements is (are) True?
$(A) \ (4, \sqrt{3}) \in S$
$(B) \ (5, \sqrt{2}) \in S$
$(C)$ Area of $R$ is $\frac{14}{3}-2 \sqrt{3}$
$(D)$ Area of $R$ is $\frac{14}{3}-\sqrt{3}$

The line $y = 2x + c$ is a tangent to the parabola $y^2 = 16x$,if $c$ equals

The equation of the parabola whose axis is vertical and passes through the points $(0, 0), (3, 0)$ and $(-1, 4)$ is

Let the line $y - \sqrt{3}x + 3 = 0$ cut the parabola $2y^2 = 2x + 3$ at $A$ and $B$. If $P(\sqrt{3}, 0)$,then the value of $|PA - PB|$ is [where $PA$ denotes the distance between points $P$ and $A$].

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 $, cm$)