In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If 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(β2x2 + α2y2) = α2β2 is.

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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 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(β²x² + α²y²) = α²β²is.

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

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

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

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

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n (E ⋃ H) = 100

n (E) + n (H) – n (E ⋂ H) = 100

75 + 40 – n (E ⋂ H) = 100

n (E ⋂ H) = 15

n (only English) = 75 – 15 = 60 = α

n (only Hindi) = 40 – 15 = 25 = β

Now ellipse 25(25²x² + 60² y² ) = (60)² × (25)²

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

eccentricity = e = $\sqrt{1 - \frac{(25)^{2}}{(60)^{2}}} = \sqrt{\frac{(85) (35)}{(60)^{2}}} = \frac{5}{60} \sqrt{17 \times 7} = \sqrt{\frac{119}{12}}$