Question

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

• A. $24$
• B. $26$
• C. $22$
• D. none of these

Answer 1

The correct option is B $26$

Let $S_1$ and $S_2$ be the foci of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $S_3$ and $S_4$ are the foci of it's conjugate hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1$

From the given information we know that length of transverse axis $=2a=4$, length of conjugate axis $=2b=6$

Hence $a=2$ and $b=3$.

Equation of hyperbola is $\frac{x^2}{4}-\frac{y^2}{9}=1$

Eccentricity of hyperbola, $e=\sqrt{\frac{a^2+b^2}{a^2}}=\frac{\sqrt{13}}{2}$

Eccentricity of conjugate hyperbola, $e_c=\sqrt{\frac{a^2+b^2}{b^2}}=\frac{\sqrt{13}}{3}$

We know that for any hyperbola the foci point are at $(ae,0)$ and $(-ae,0)$ and for any conjugate hyperbola the foci are at $(0,b{e}_c)$ and $(0,-b{e}_c)$

Hence $S_1$ is $(2\times \frac{\sqrt{13}}{2},0)$ and $S_2$ is $(-2\times \frac{\sqrt{13}}{2},0)$

or $S_1$ is $(\sqrt{13},0)$ and $S_2$ is $(-\sqrt{13},0)$

Similarly $S_3$ is $(0,3\times \frac{\sqrt{13}}{3})$ and $S_4$ is $(0,-3\times \frac{\sqrt{13}}{3})$

or $S_3$ is $(0,\sqrt{13})$ and $S_4$ is $(0,-\sqrt{13})$

We can see from these results that distance of all four vertices of quadrilateral is same and equal to $\sqrt{13}$, also the diagonals are at right angle.

hence the quadrilateral is a square with side length equal to $S_1{S}_3=S_3{S}_2=S_2{S}_4=S_4{S}_1=\sqrt{26}$

Area of square $S_1{S}_3{S}_2{S}_4$ is $(\sqrt{26}{)}^2$

$\to$ Area $=26$