If the tangents of the parabola $y^2=8x$ passing through the point $P(1,3)$ touch the parabola at $A$ and $B$,then the area (in sq. units) of $\triangle PAB$ is

Let $R$ be the focus of the parabola $y^2=20x$ and the line $y=mx+c$ intersect the parabola at two points $P$ and $Q$. Let the point $G(10, 10)$ be the centroid of the triangle $PQR$. If $c-m=6$,then $(PQ)^2$ is

If $(a, b)$ is the midpoint of a chord passing through the vertex of the parabola $y^2 = 4x$,then:

$A$ tangent to the parabola $y^2 = 8x$ makes an angle of $45^\circ$ with the straight line $y = 3x + 5$. Find the equation of the tangent.

Let $S$ be the set of all $a \in \mathbb{N}$ such that the area of the triangle formed by the tangent at the point $P(b, c)$,where $b, c \in \mathbb{N}$,on the parabola $y^2 = 2ax$ and the lines $x = b$ and $y = 0$ is $16 \text{ unit}^2$. Then $\sum_{a \in S} a$ is equal to $..........$.

Find the coordinates of the focus,axis of the parabola,the equation of the directrix,and the length of the latus rectum for $y^{2}=10x$.