If the length of the latus rectum of a parabola,whose focus is $(a, a)$ and the tangent at its vertex is $x+y=a$,is $16$,then $|a|$ is equal to.

$A$ line passing through the point of intersection of $x+y=4$ and $x-y=2$ makes an angle $\tan^{-1}\left(\frac{3}{4}\right)$ with the $X$-axis. It intersects the parabola $y^{2}=4(x-3)$ at points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$, respectively. Then $|x_{1}-x_{2}|$ is equal to

Consider the parabola with vertex $\left(\frac{1}{2}, \frac{3}{4}\right)$ and the directrix $y=\frac{1}{2}$. Let $P$ be the point where the parabola meets the line $x=-\frac{1}{2}$. If the normal to the parabola at $P$ intersects the parabola again at the point $Q$,then $(PQ)^{2}$ is equal to :

If the line $x = my + k$ touches the parabola $x^2 = 4ay$,then $k = $

The equation of the tangent to the parabola $y^{2}=4x$ inclined at an angle of $\frac{\pi}{4}$ to the positive direction of $x$-axis is:

If the $x$-intercept of a focal chord of the parabola $y^2 = 8x + 4y + 4$ is $3$,then the length of this chord is equal to $.............$