JEE Main 2023 — Sets and Relations Question with Solution

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,

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

$\Rightarrow \frac{x^2}{36\times 4}+\frac{y^2}{25}=1$

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

Now we know that, 

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

$\Rightarrow e=\sqrt{1-\frac{25}{36\times 4}}=\frac{\sqrt{119}}{12}$