Correct Answer - Option 3 :
\(\frac{8\sqrt2}{3}\) units
Concept:
Equation of ellipse: \( \rm \frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 \)
Length of latus rectum = \(\rm \frac{2b^2}{a}\), when a > b
Calculation:
Given: equation of ellipse \(\rm 8x^2+18y^2=144\)
This may be written as, \( \rm \frac{x^2}{18}+ \frac{y^2}{8}=1 \)...........(divide by 144)
\(\Rightarrow \rm \frac{x^2}{(3√2)^2}+ \frac{y^2}{(2√2)^2}=1 \)
This is of form \( \rm \frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 \), with a > b
Here, a = 3√2 and b = 2√2
Now, we know the length of latus rectum = \(\rm \frac{2b^2}{a}\)
\(=\frac{2(2\sqrt2)^2}{3\sqrt2}\)
\(=\frac{8\sqrt2}{3}\) units.
Hence, option (3) is correct.
