Find the length of the chord intercepted by the line $4y = 3x - 48$ on the parabola $y^{2} = 64x$.

The slope of the normal at the point $(at^2, 2at)$ of the parabola $y^2 = 4ax$ is

The maximum area of a circle centered at the origin,which is inscribed in the parabola $y = x^2 - 100$,can be expressed as $\frac{a\pi}{b}$,where $a$ and $b$ are coprime numbers. Then the value of $a + b$ 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:

Find the equation of the common tangent to the parabolas $y^2 = 2x$ and $x^2 = 16y$.

If $(h, k)$ is the point to which the origin is shifted in order to transform the equation $y^2-4x+6y+17=0$ into the form $Y^2=4aX$,then $h^2+k^2=$