In a group of 100 persons, 75 speak English and 40 speak Hindi. E… — Solution | JEE Mathematics

MCQMediumJEE 2023Conic Sections (Parabola, Ellipse, Hyperbola)

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

A
$\frac{\sqrt{129}}{12}$
B
$\frac{\sqrt{117}}{12}$
C
$\frac{\sqrt{119}}{12}$
D
$\frac{\sqrt{15}}{12}$

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: In a group of $100$ persons, $75$ speak English and $40$ speak Hindi, and each person speaks at least one language. The number speaking only English is $\alpha$ and only Hindi is $\beta$ .

Find: The eccentricity of the ellipse $25\left(\beta^2 x^2 + \alpha^2 y^2 \right) = \alpha \beta^2$ .

From the solution, we use $\alpha = 60$ and $\beta = 25$ .

For the ellipse, the eccentricity is computed as

e^2 = 1 - \frac{25 \times 25}{60^2} = \frac{119}{144}

Concluding Value

Taking square root,

e = \frac{\sqrt{119}}{12}

Therefore, the eccentricity of the ellipse is $\frac{\sqrt{119}}{12}$ .

The working shown in the solution leads to option C. However, the provided correct answer field marks B as correct.

Common mistakes

Using only the counts $75$ and $40$ directly as $\alpha$ and $\beta$ is incorrect, because $\alpha$ and $\beta$ represent only English and only Hindi, not total speakers of those languages. First separate the overlap, then identify the exclusive counts.
Confusing the eccentricity formula for an ellipse is a common mistake. One must first identify the larger denominator as $a^2$ and then use $e = \sqrt{1 - \frac{b^2}{a^2}}$ . Reversing $a^2$ and $b^2$ gives a wrong value.
Trusting the option label without checking the computed value can cause an error here. The solution computes $\frac{\sqrt{119}}{12}$ , so always verify the derived expression against the listed options before final selection.

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Answer

Step-by-step solution

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