Question

If $S_1$ and $S_2$ are the foci of the hyperbola whose transverse axis length is $4$ units and conjugate axis length is $6$ units, $S_3$ and $S_4$ are the foci of the conjugate hyperbola, then the area of the quadrilateral $S_1{S}_3{S}_2{S}_4$ is sq. units

Answer 1

Since, we know that foci of the hyperbola and its conjugate hyperbola are concyclic and form a square.


Let $e$ and $e_1$ be the eccentricities of hyperbola and its conjugate hyperbola respetively.
$\therefore$ Required area $=4\text{Area}\Delta S_{2}O{S}_4=4\times \frac{1}{2}ae\times b{e}_1=4\times \frac{1}{2}\times 2\times 3\times e{e}_1$
$\because b^2=a^2(e^2-1)\Rightarrow e^2=\frac{9}{4}+1=\frac{13}{4}$
Also $\frac{1}{e_1^2}=1-\frac{1}{e^2}=1-\frac{4}{13}=\frac{9}{13}$
$e_1^2=\frac{13}{9}$
Required area $=12\times \frac{\sqrt{13}}{2}\times \frac{\sqrt{13}}{3}$
$=2\times 13=26$ sq. units