Correct Answer - Option 1 :
\(\frac{\sqrt{15}}{3}\)
Concept:
Equation of hyperbola, \(\rm \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1\)
Eccentricity, e = \(\rm\sqrt{1+\frac{b^{2}}{a^{2}}}\)
Equation of hyperbola, \(\rm -\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}= 1\)
Eccentricity, e = \(\rm\sqrt{1+\frac{a^{2}}{b^{2}}}\)
Calculation:
Given equation of hyperbola , 2x2 - 3y2 = 18
⇒ \(\rm \frac{x^{2}}{9}- \frac{y^{2}}{6}=1\)
on comparing with standard equation ,\(\rm \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1\)
we have , a = 3 and b = \(\sqrt{6}\)
We know that eccentricity, e = \(\rm\sqrt{1+\frac{b^{2}}{a^{2}}}\)
⇒ e = \(\sqrt{1+\frac{6}{9}}\)
⇒ e = \(\frac{\sqrt{15}}{3}\) .
The correct option is 1.