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Find the eccentricity of hyperbola 2x2 - 3y2 = 18 .
1. \(\frac{\sqrt{15}}{3}\)
2. \(\frac{\sqrt{15}}{9}\)
3. \(\frac{\sqrt{5}}{3}\)
4. \({\sqrt{3}}{}\)

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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. 

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