नाभियाँ $(0,±3)$ और शीर्षों $\left(0, \pm \frac{\sqrt{11}}{2}\right)$ वाले अतिपरवलय का समीकरण ज्ञात
कीजिए।

Solution since the foci is on $y-$ axis, the equation of the hyperbola is of the form $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$

since vertices are $\left(0,\,\pm \frac{\sqrt{11}}{2}\right)$ ,   $a=\frac{\sqrt{11}}{2}$

Also, since foci are $(0,\,±3)$;   $c=3$ and $b^{2}=c^{2}-a^{2}=\frac{25}{4}$

Therefore, the equation of the hyperbola is

$\frac{y^{2}}{\left(\frac{11}{4}\right)}$ $-\frac{x^{2}}{\left(\frac{25}{4}\right)}=1$, i.e., $100 y^{2}-44 x^{2}=275$