Correct Answer - Option 3 :
\(\sqrt { \frac{7}{4}}\)
Concept:
Standard equation of an hyperbola : \(\frac{{{\rm{\;}}{{\bf{x}}^2}}}{{{{\bf{a}}^2}}} - \frac{{{{\bf{y}}^2}}}{{{{\bf{b}}^2}}} = 1\)
- Coordinates of foci = (± ae, 0)
- Eccentricity (e) = \(\sqrt {1 + {\rm{\;}}\frac{{{{\rm{b}}^2}}}{{{{\rm{a}}^2}}}} \) ⇔ a2e2 = a2 + b2
- Length of Latus rectum = \(\rm \frac{2b^2}{a}\)
Calculation:
Given: \(\rm \frac{x^2}{100} - \frac{y^2}{75} = 1\)
Compare with the standard equation of a hyperbola: \(\frac{{{\rm{\;}}{{\bf{x}}^2}}}{{{{\bf{a}}^2}}} - \frac{{{{\bf{y}}^2}}}{{{{\bf{b}}^2}}} = 1\)
So, a2 = 100 and b2 = 75
Now, Eccentricity (e) = \(\sqrt {1 + {\rm{\;}}\frac{{{{\rm{b}}^2}}}{{{{\rm{a}}^2}}}} \)
= \(\sqrt {1 + \frac{75}{100}}\)
= \(\sqrt {1 + \frac{3}{4}}\)
= \(\sqrt { \frac{7}{4}}\)