Correct Answer - Option 4 :
\(\frac{2\sqrt2}{3} \) units
Concept:
Equation of ellipse: \(\rm \frac{x^2}{a^2}+ \frac{y^2}{b^2}=1\)
Eccentricity:
- \( \rm\sqrt{1- \frac{b^2}{a^2}}\) when a > b
-
\( \rm\sqrt{1- \frac{a^2}{b^2}}\) when a < b
Calculation:
Given: equation of ellipse, \(\rm 36x^2+4y^2=144\)
This may be written as \(\rm \frac{x^2}{4}+\rm \frac{y^2}{36}=1\)...........(divide by 144)
Here, \(\rm a^2=4, b^2=36\)
This is of the form \(\rm \frac{x^2}{a^2}+ \frac{y^2}{b^2}=1\), where, \(\rm a^2<b^2\)
∴ Eccentricity = \( \rm\sqrt{1- \frac{a^2}{b^2}}=\sqrt{1-\frac{4}{36}}\)
\(=\sqrt{\frac{32}{36}} \)
\(=\frac{4\sqrt2}{6} \)
\(=\frac{2\sqrt2}{3} \) units
Hence, option (4) is correct.
