Correct Answer - Option 4 :
\(\dfrac{\sqrt{13}}{2}\)
CONCEPT:
The eccentricity of a hyperbola of the form \(\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1\) is given by: \(e = \sqrt {1 + \frac{{{b^2}}}{{{a^2}}}} \)
CALCULATION:
Given: The point (3tan θ, 2sec θ) lies on the hyperbola.
As we can see that, if we substitute x = 3tan θ and y = 2sec θ in the expression \(\frac{y^2}{4}-\frac{x^2}{9} \) we get
⇒ \(\frac{y^2}{4}-\frac{x^2}{9} = sec^2 \ \theta - tan^2 \ \theta = 1\)
So, we can say that, the point (3tan θ, 2sec θ) lies on the hyperbola \(\frac{y^2}{4}-\frac{x^2}{9} = 1\)
As we know that, eccentricity of the hyperbola of the form \(\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1\) is given by: \(e = \sqrt {1 + \frac{{{b^2}}}{{{a^2}}}} \)
Here, a2 = 4 and b2 = 9
⇒ \(e = \sqrt {1 + \frac{{{9}}}{{{4}}}} = \frac{\sqrt {13}}{2} \)
Hence, correct option is 4.