Let an ellipse \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a [JEE Main 2026, 2 Apr (Shift 1)]

Given \( \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \)

passes \((4,3) \)
\( \frac{16}{a^2}+\frac{9}{b^2}=1 \ldots \)(1)
\(e=\frac{\sqrt{5}}{3} \)
\( e^2=\frac{5}{9}\)
\( 1-\frac{a^2}{b^2}=\frac{5}{9} \)
\( \frac{a^2}{b^2}=\frac{4}{9} \ldots\)(2)

From (1) and (2)

\( \frac{16}{a^2}+\frac{4}{a^2}=1\)
\( \frac{20}{a^2}=1 \Rightarrow a^2=20 \ \& \ b^2=45\)
\(\ell(L R)=\frac{2 a^2}{b}=\frac{2 \times 20}{\sqrt{45}}=\frac{40}{\sqrt{45}}\)
\( =\frac{40 \sqrt{5}}{3 \sqrt{5} \times \sqrt{5}}=\frac{8 \sqrt{5}}{3}\)