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In a group of $100$ persons $75$ speak English and $40$ speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $\alpha$ and the number of persons who speak only Hindi is $\beta$, then the eccentricity of the ellipse $25\left(\beta^2 x^2+\alpha^2 y^2\right)=\alpha^2 \beta^2$ is $.......$
$\frac{3 \sqrt{15}}{12}$
$\frac{\sqrt{117}}{12}$
$\frac{\sqrt{119}}{12}$
$\frac{\sqrt{129}}{12}$
Solution
$\alpha+ p =75$
$\beta+ p =40$
$\alpha+\beta+ p =100$
$\text { From }(1),(2) \text { and (3) }$
$P =15, \alpha=60 \text { and } \beta=25$
$\text { Now equation of ellipse: } 25\left(\frac{ x ^2}{\alpha^2}+\frac{ y ^2}{\beta^2}\right)=1$
$\frac{ x ^2}{144}+\frac{ y ^2}{25}=1$
$\Rightarrow e=\frac{\sqrt{119}}{12}$
