Which of the following equations in parametric form can represent a hyperbola, where $'t'$  is a parameter.

The equation of the hyperbola in the standard form (with transverse axis along the $x$ –  axis) having the length of the latus rectum = $9$ units and eccentricity = $5/4$ is

Let the focal chord of the parabola $P: y^{2}=4 x$ along the line $L: y=m x+c, m>0$ meet the parabola at the points $M$ and $N$. Let the line $L$ be a tangent to the hyperbola $H : x ^{2}- y ^{2}=4$. If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive $x$-axis, then the area of the quadrilateral $OMFN$ is.

If $\frac{{{{\left( {3x – 4y – z} \right)}^2}}}{{100}} – {\frac{{\left( {4x + 3y – 1} \right)}}{{225}}^2} = 1$ then
length of latusrectum of hyperbola is

The locus of the middle points of the chords of hyperbola $3{x^2} – 2{y^2} + 4x – 6y = 0$ parallel to $y = 2x$ is