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 and the number of persons who speak only Hindi is , then the eccentricity of the elli

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Solution

Q.
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 $α$ and the number of persons who speak only Hindi is $β$ , then the eccentricity of the ellipse $25 (β^{2} x^{2} + α^{2} y^{2}) = α^{2} β^{2}$ is [2023]

1 $\frac{\sqrt{117}}{12}$

2 $\frac{\sqrt{119}}{12}$

3 $\frac{\sqrt{129}}{12}$

4 $\frac{3 \sqrt{15}}{12}$

Ans.
(2)

Let $γ$ be the number of persons who speak both English and Hindi.

According to the question, we have

   $α + β + γ = 100 (i)$

   $α + γ = 75 (ii)$

and    $β + γ = 40 (iii)$

Solving (i), (ii) and (iii), we get

$α = 60,$ $β = 25$ and $γ = 15$

So, equation of the given ellipse becomes,

$25 (25^{2} x^{2} + 60^{2} y^{2}) = 60^{2} \times 25^{2}$

$\Rightarrow \frac{x^{2}}{{(60 / 5)}^{2}} + \frac{y^{2}}{{(25 / 5)}^{2}} = 1 \Rightarrow \frac{x^{2}}{12^{2}} + \frac{y^{2}}{5^{2}} = 1$

So, eccentricity of ellipse $= \sqrt{1 - \frac{b^{2}}{a^{2}}} = \sqrt{1 - \frac{5^{2}}{12^{2}}} = \frac{\sqrt{119}}{12}$

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