If one end of a focal chord of the parabola $y^2=\frac{8}{a} \times(a>0)$ is at $(1,4)$, then the length of this focal chord is

If $m$ is the length of the latusrectum and $n$ is the length of the major-axis of the ellipse $25 x^2+16 y^2-150 x-64 y-111=0$, then the ordered pair $(m, n)=$

If $P(\theta)$ and $Q\left(\frac{\pi}{2}+\theta\right)$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the locus of mid-point of $P Q$ is $\frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}=1$, then $\frac{a+b}{\alpha+\beta}=$

Let $S$ be the focus of the hyperbola $x^2-2 y^2=1$ lying on the positive $X$-axis. Let $P(-1,1)$ be a given point. Then, the area of the triangle formed by the line $P S$ with the coordinate axes is (in sq. units)