The length of the chord of the parabola $x^2 = 4y$ having the equation $x - \sqrt{2}y + 4\sqrt{2} = 0$ is

The area of the triangle formed by the lines joining the vertex of the parabola $x^2 = 12y$ to the ends of its latus rectum is .................. $sq. \ unit$.

The shortest distance between the line $y = x$ and the curve $y^2 = x - 2$ is

The coordinates of a point on the parabola $y^2 = 8x$ whose focal distance is $4$ are

Let $P$ be a point on the parabola $y^2 = 4ax$,where $a > 0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $PFQ$,where $F$ is the focus of the parabola,is $120$. If the slope $m$ of the normal and $a$ are both positive integers,then the pair $(a, m)$ is

Match the items given in List-$A$ with those of the items of List-$B$:

List-$A$	List-$B$
$(A)$. The vertex of the parabola $y^2+4x-2y+3=0$ is	$(I)$. $\left(\frac{5}{4}, 1\right)$
$(B)$. The vertex of the parabola $x^2+8x+12y+4=0$ is	$(II)$. $\left(1, \frac{5}{4}\right)$
$(C)$. The focus of the parabola $y^2-x-2y+2=0$ is	$(III)$. $\left(-\frac{1}{2}, 1\right)$
$(D)$. The focus of the parabola $x^2-2x-8y-23=0$ is	$(IV)$. $(1, -1)$
	$(V)$. $(-4, 1)$

The correct match is: