JEE Main 2023 — Mathematics Algebra

Given,

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

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

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

Now from above diagram we can see that,

$$\beta =100-75=25$$β=100−75=25

So, $\alpha =75-(40-25)=60$α=75−(40−25)=60

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

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

$$25[\frac{{x}^{2}}{{60}^{2}}+\frac{{y}^{2}}{{25}^{2}}]=1$$25[602x2​+252y2​]=1

$$\Rightarrow \frac{{x}^{2}}{36\times 4}+\frac{{y}^{2}}{25}=1$$⇒36×4x2​+25y2​=1

Now comparing with $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$a2x2​+b2y2​=1 we get, a=12&b=5

Now we know that,

Eccentricity of ellipse is given by $e=\sqrt{1-\frac{{b}^{2}}{{a}^{2}}}$e=1−a2b2​​

$$\Rightarrow e=\sqrt{1-\frac{25}{36\times 4}}=\frac{\sqrt{119}}{12}$$⇒e=1−36×425​​=12119​​