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

Question

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

Answer: $e = \dfrac{\sqrt{119}}{12}$ (Option C in the source)

Accepted Answer

$n(E\cup H)=100$, so $n(E\cap H) = 75+40-100 = 15$.

$\alpha = 75-15 = 60$ (only English), $\beta = 40-15 = 25$ (only Hindi).

Rewrite the ellipse: dividing $25\beta^2x^2+25\alpha^2y^2=\alpha^2\beta^2$ by $\alpha^2\beta^2$:
$$\frac{x^2}{\alpha^2/25} + \frac{y^2}{\beta^2/25} = 1$$
So $a^2 = \dfrac{\alpha^2}{25} = \dfrac{3600}{25}=144$, $b^2=\dfrac{\beta^2}{25}=\dfrac{625}{25}=25$. Since $a^2>b^2$ (major axis along $x$):
$$e = \sqrt{1-\frac{b^2}{a^2}} = \sqrt{1-\frac{25}{144}} = \sqrt{\frac{119}{144}} = \frac{\sqrt{119}}{12}$$

*(The exact answer-choice wording was not recoverable from the source text, but the answer is marked as option C, matching this computed eccentricity.)*

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