In a group of 100 persons 75 speak English and 40 speak Hindi. Each...

Given,

In a group of $100$ persons $75$ speak English and $40$ speak Hindi,

And each person speaks at least one of the two languages,

And the number of persons who speak only English is $\alpha$ and the number of persons who speaks only Hindi is $\beta$,

Now from above diagram we can see that,

$\beta = 100 - 75 = 25$

So, $\alpha = 75 - \left(40 - 25\right) = 60$

Hence, number of people who speak English only is $\alpha = 60$ and Hindi is $\beta = 25$,

Now putting the value in given equation of ellipse we get,

$25\left[\dfrac{{x}^{2}}{{60}^{2}} + \dfrac{{y}^{2}}{{25}^{2}}\right] = 1$

$⇒ \dfrac{{x}^{2}}{36 \times 4} + \dfrac{{y}^{2}}{25} = 1$

Now comparing with $\dfrac{{x}^{2}}{{a}^{2}} + \dfrac{{y}^{2}}{{b}^{2}} = 1$ we get, $a = 12 \text{ and } b = 5$

Now we know that,

Eccentricity of ellipse is given by $e = \sqrt{1 - \dfrac{{b}^{2}}{{a}^{2}}}$

$⇒ e = \sqrt{1 - \dfrac{25}{36 \times 4}} = \dfrac{\sqrt{119}}{12}$