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 speaks only Hindi is $$\beta$$, then the eccentricity of the ellipse $$25(\beta^2 x^2 + \alpha^2 y^2) = \alpha^2\beta^2$$ is

A total of 100 persons includes 75 English speakers and 40 Hindi speakers, each speaking at least one language. By the inclusion-exclusion principle, the number of persons speaking both languages is 75 + 40 - 100 = 15, so the number speaking only English is $$\alpha$$ = 75 - 15 = 60 and the number speaking only Hindi is $$\beta$$ = 40 - 15 = 25.

The equation of the ellipse can be written as $$25(\beta^2 x^2 + \alpha^2 y^2) = \alpha^2\beta^2$$
which yields $$\frac{x^2}{\alpha^2/25} + \frac{y^2}{\beta^2/25} = 1 \implies \frac{x^2}{144} + \frac{y^2}{25} = 1\ .$$

In this form, $$a^2 = 144$$ and $$b^2 = 25$$ since $$\alpha^2/25 = 3600/25 = 144$$ and $$\beta^2/25 = 625/25 = 25$$.

The eccentricity of the ellipse is given by $$ e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{25}{144}} = \sqrt{\frac{119}{144}} = \frac{\sqrt{119}}{12}\ .$$

The correct answer is $$\dfrac{\sqrt{119}}{12}$$.