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 a and the number of persons who speak only Hindi is b, then the eccentricity of the ellipse 25(β2 x² + α² y2) = α² β² is

• A. \(\frac{117}{12}\)
• B. \(\frac{119}{12}\)
• C. \(\frac{129}{12}\)
• D. \(3\frac{15}{12}\)

Answer 1

The Correct Option is B

To solve this problem, we need to tackle it in two parts. The first part involves determining the values of \(a\) (those who speak only English) and \(b\) (those who speak only Hindi) based on the given language statistics. The second part is solving the equation of the ellipse to find its eccentricity.

Part 1: Language Statistics

In a group of 100 persons, the statistics are provided as follows:

• Total persons = 100
• Persons speaking English = 75
• Persons speaking Hindi = 40
• Each person speaks at least one of the two languages.

Using the principle of inclusion-exclusion for sets, we can calculate as follows:

\(n(\text{Only English}) = n(\text{English}) - n(\text{Both})\)  
\(n(\text{Only Hindi}) = n(\text{Hindi}) - n(\text{Both})\) 
\(n(\text{Both}) = n(\text{English}) + n(\text{Hindi}) - n(\text{Union})\)

Substituting the known values:

\(n(\text{Both}) = 75 + 40 - 100 = 15\)

Therefore:

\(a = n(\text{Only English}) = 75 - 15 = 60\) 
\(b = n(\text{Only Hindi}) = 40 - 15 = 25\)

Part 2: Eccentricity of the Ellipse

We have the equation of the ellipse: \(25(\beta^2 x^2 + \alpha^2 y^2) = \alpha^2 \beta^2\). To find the eccentricity, we first express this in standard form:

\(\frac{x^2}{\left(\frac{\alpha^2 \beta^2}{25 \beta^2}\right)} + \frac{y^2}{\left(\frac{\alpha^2 \beta^2}{25 \alpha^2}\right)} = 1\)

Thus, the semi-major axis (\(a\)) and the semi-minor axis (\(b\)) are given by:

\(a^2 = \frac{\alpha^2 \beta^2}{25 \beta^2} = \frac{\alpha^2}{25}\), \(b^2 = \frac{\alpha^2 \beta^2}{25 \alpha^2} = \frac{\beta^2}{25}\)

The eccentricity \(e\) is given by the formula:

\(e = \sqrt{1 - \frac{b^2}{a^2}}\)

Substituting the values:

\(e = \sqrt{1 - \frac{\frac{\beta^2}{25}}{\frac{\alpha^2}{25}}} = \sqrt{1 - \frac{\beta^2}{\alpha^2}}\)

Plugging in the values for \(a = 60\) and \(b = 25\) based on the language problem:

\(e = \sqrt{1 - \left(\frac{25}{60}\right)^2} = \sqrt{1 - \left(\frac{5}{12}\right)^2}\)

After calculating, the eccentricity of the ellipse is found to be:

\(\frac{119}{12}\)

The correct option matches with the derived eccentricity value, so the answer is correct.