If the vertex and focus of a parabola lying on the $x$-axis are at distances $p$ and $q$ from the origin respectively,find its equation.

Suppose a parabola passes through $(0,4), (1,9)$ and $(4,5)$ and has its axis parallel to the $y$-axis. Then the equation of the parabola is

For the parabola $y=x^2-3x+2$,match the items in List-$I$ to that of the items in List-$II$. $S$ is a focus,$Z$ is the intersection of the axis and the directrix,$P$ is one end point of the latus rectum,$Q$ is the point on the parabola at which the tangent is parallel to the $X$-axis.

$A$. $P$	$I$. $(2,0)$
$B$. $Q$	$II$. $(\frac{3}{2}, -\frac{1}{4})$
$C$. $S$	$III$. $(\frac{3}{2}, 0)$
$D$. $Z$	$IV$. $(\frac{3}{2}, -\frac{1}{2})$
	$V$. $(0, \frac{3}{2})$

If $L$ is the normal drawn to the parabola $y^2 = 8x$ at the point $t = \frac{1}{\sqrt{2}}$,then the foot of the perpendicular drawn from the focus of the parabola on to the normal $L$ is

Let $P : y^{2} = 4ax, a > 0$ be a parabola with focus $S$. Let the tangents to the parabola $P$ that make an angle of $\frac{\pi}{4}$ with the line $y = 3x + 5$ touch the parabola $P$ at $A$ and $B$. Then the value of $a$ for which $A, B$ and $S$ are collinear is:

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