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 $25\left(\beta^{2} \mathrm{x}^{2}+\alpha^{2} \mathrm{y}^{2}\right)=\alpha^{2} \beta^{2}$ is

JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Ellipse

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}

(A) $\frac{3 15}{12}$
(B) $\frac{117}{12}$
(C) $\frac{119}{12}$ ✓ Correct
(D) $\frac{129}{12}$

Answer: (C)

Solution

Let p be the number speaking both languages. Then

α + p = 75

β + p = 40

, and

α + β + p = 100

. Solving gives

p = 15

α = 60

β = 25

. The ellipse equation becomes

\frac{x ^{2}}{α ^{2} /25} + \frac{y ^{2}}{β ^{2} /25} = 1

, i.e.,

\frac{x ^{2}}{144} + \frac{y ^{2}}{25} = 1

. Since

a^{2} = 144

and

b^{2} = 25

, we have

e = 1 - \frac{25}{144} = \frac{119}{144} = \frac{119}{12}

JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Single correct option

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JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Ellipse

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