If $P(\frac{\pi}{4})$ and $Q(\frac{3 \pi}{4})$ are two points on the hyperbola $4 x^2-y^2-8 x-2 y-13=0$ in parametric form,then the distance between $P$ and $Q$ is

The line $y = mx + c$ touches the curve $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,if

Let $R$ be a rectangle given by the lines $x=0, x=2, y=0$ and $y=5$. Let $A(\alpha, 0)$ and $B(0, \beta)$,where $\alpha \in [0, 2]$ and $\beta \in [0, 5]$,be such that the line segment $AB$ divides the area of the rectangle $R$ in the ratio $4:1$. Then,the mid-point of $AB$ lies on a $.........$.

If the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ has eccentricity $e = \frac{5}{4}$ and the length of the latus rectum equal to $9$,then $ab$ is equal to

The eccentricity of the hyperbola $4x^2 - 9y^2 = 16$ is

If a hyperbola has asymptotes $3x - 4y - 1 = 0$ and $4x - 3y - 6 = 0$,then the transverse and conjugate axes of that hyperbola are