This problem is a beautiful blend of Set Theory and Coordinate Geometry, specifically involving Ellipses. Our goal is to first extract numerical values for $\alpha$ and $\beta$ using the given information about language speakers, and then use these values to determine the eccentricity of a given ellipse.

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Part 1: Determining the Values of $\alpha$ and $\beta$ (Using Set Theory)

The first part of the problem requires us to interpret information about groups of people speaking languages. This is a classic application of set theory principles, particularly the Principle of Inclusion-Exclusion.

Key Concepts: Set Theory and the Principle of Inclusion-Exclusion

For any two finite sets, $A$ and $B$:

1. Number of elements in the Union: The number of elements in the union of sets $A$ and $B$, denoted $n(A \cup B)$, is given by: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ Here, $n(A \cap B)$ represents the number of elements common to both sets $A$ and $B$ (the intersection). We subtract $n(A \cap B)$ because elements in the intersection are counted twice (once in $n(A)$ and once in $n(B)$).

2. Number of elements "Only in A": The number of elements that are in set $A$ but not in set $B$ (often written as $n(A \setminus B)$ or $n(A \text{ only})$) is: $n(A \setminus B) = n(A) - n(A \cap B)$ Similarly, the number of elements "Only in B" is $n(B \setminus A) = n(B) - n(A \cap B)$.

Step-by-Step Calculation of $\alpha$ and $\beta$

Step 1: Define Sets and State Given Information. Let's clearly define our sets based on the problem statement:

• Let $U$ be the universal set representing all persons in the group. We are given $n(U) = 100$.
• Let $E$ be the set of persons who speak English. We are given $n(E) = 75$.
• Let $H$ be the set of persons who speak Hindi. We are given $n(H) = 40$.

The problem states, "Each person speaks at least one of the two languages." This is a crucial piece of information. It implies that there are no persons who speak neither English nor Hindi. Therefore, the total number of persons in the group is equal to the number of persons who speak English or Hindi (or both). $n(E \cup H) = n(U) = 100$

Step 2: Calculate the Number of Persons Who Speak Both Languages ($n(E \cap H)$). To find the number of people who speak only English ($\alpha$) and only Hindi ($\beta$), we first need to determine the number of people who speak both languages. This is the common part that gets subtracted from the total speakers of each language. Using the Principle of Inclusion-Exclusion formula: $n(E \cup H) = n(E) + n(H) - n(E \cap H)$ Substitute the known values into this equation: $100 = 75 + 40 - n(E \cap H)$ $100 = 115 - n(E \cap H)$ Now, we solve for $n(E \cap H)$: $n(E \cap H) = 115 - 100$ $n(E \cap H) = 15$ So, 15 persons speak both English and Hindi.

Step 3: Calculate $\alpha$, the Number of Persons Who Speak Only English. The problem defines $\alpha$ as the number of persons who speak only English. This means they are in set $E$ but not in set $H$. Using the formula for "only A": $\alpha = n(E \setminus H) = n(E) - n(E \cap H)$ Substitute the values we have: $\alpha = 75 - 15$ $\alpha = 60$ Thus, 60 persons speak only English.

Step 4: Calculate $\beta$, the Number of Persons Who Speak Only Hindi. Similarly, $\beta$ is defined as the number of persons who speak only Hindi. This means they are in set $H$ but not in set $E$. Using the formula for "only B": $\beta = n(H \setminus E) = n(H) - n(E \cap H)$ Substitute the values: $\beta = 40 - 15$ $\beta = 25$ Hence, 25 persons speak only Hindi.

Educational Tip: Visualizing with a Venn Diagram It's always a good practice to visualize these problems with a Venn diagram to ensure your calculations are consistent.

• Draw two overlapping circles, one for English (E) and one for Hindi (H).
• Start by filling the intersection: $n(E \cap H) = 15$.
• Then, fill the "only English" region: $n(E) - n(E \cap H) = 75 - 15 = 60$ (which is $\alpha$).
• Next, fill the "only Hindi" region: $n(H) - n(E \cap H) = 40 - 15 = 25$ (which is $\beta$).
• Finally, check the total: $60 \text{ (only E)} + 25 \text{ (only H)} + 15 \text{ (both)} = 100$. This matches $n(E \cup H)$, confirming our values for $\alpha$ and $\beta$ are correct.

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Part 2: Analyzing the Ellipse Equation and Calculating Eccentricity

Now that we have the values $\alpha = 60$ and $\beta = 25$, we will use them in the given ellipse equation to find its eccentricity.

Key Concepts: Standard Equation and Eccentricity of an Ellipse

An ellipse centered at the origin has one of two standard forms:

1. Major axis along the x-axis (horizontal ellipse): $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \text{where } a > b$ In this case, $a$ is the length of the semi-major axis (half the length of the longer axis) and $b$ is the length of the semi-minor axis (half the length of the shorter axis).

2. Major axis along the y-axis (vertical ellipse): $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \quad \text{where } a > b$ Again, $a$ is the semi-major axis and $b$ is the semi-minor axis. Note that $a$ always represents the semi-major axis (the larger one), and $b$ always represents the semi-minor axis (the smaller one).

Eccentricity ($e$) of an Ellipse: Eccentricity is a measure of how "stretched out" or "circular" an ellipse is. For an ellipse, $0 < e < 1$. The formula for eccentricity is derived from the relationship between the semi-major axis ($a$), semi-minor axis ($b$), and the distance from the center to each focus ($c$), where $c^2 = a^2 - b^2$. Eccentricity is defined as $e = c/a$. Substituting $c = \sqrt{a^2 - b^2}$, we get: $e = \frac{\sqrt{a^2 - b^2}}{a} = \sqrt{\frac{a^2 - b^2}{a^2}} = \sqrt{1 - \frac{b^2}{a^2}}$ This formula is general. Always remember: $a$ is the semi-major axis and $b$ is the semi-minor axis. So, the formula is $e = \sqrt{1 - \frac{(\text{semi-minor axis})^2}{(\text{semi-major axis})^2}}$.

Step-by-Step Calculation of Eccentricity

Step 5: Substitute $\alpha$ and $\beta$ into the Ellipse Equation. The given equation of the ellipse is: $25\left(\beta^{2} x^{2}+\alpha^{2} y^{2}\right)=\alpha^{2} \beta^{2}$ Now, substitute the calculated values $\alpha = 60$ and $\beta = 25$ into this equation. This is our first step to convert the general form into a specific numerical equation. $25\left((25)^{2} x^{2}+(60)^{2} y^{2}\right)=(60)^{2} (25)^{2}$ $25\left(625 x^{2}+3600 y^{2}\right)=3600 \times 625$

Step 6: Convert the Equation to Standard Form. To find the eccentricity, we need the equation in its standard form $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$. To achieve this, we must make the right-hand side of the equation equal to 1. We do this by dividing both sides of the equation by the term on the right-hand side, which is $(60)^2 (25)^2 = 3600 \times 625$. $\frac{25\left(625 x^{2}+3600 y^{2}\right)}{3600 \times 625} = \frac{3600 \times 625}{3600 \times 625}$ $\frac{25 \times 625 x^{2}}{3600 \times 625} + \frac{25 \times 3600 y^{2}}{3600 \times 625} = 1$ Now, we simplify the coefficients by canceling common factors. Notice that $625 = 25^2$. $\frac{25 x^{2}}{3600} + \frac{25 y^{2}}{625} = 1$ To get $x^2$ and $y^2$ with coefficients of 1, we move their numerical coefficients to the denominator: $\frac{x^{2}}{\frac{3600}{25}} + \frac{y^{2}}{\frac{625}{25}} = 1$ Perform the divisions: $\frac{x^{2}}{144} + \frac{y^{2}}{25} = 1$ This is the standard form of our ellipse equation.

Step 7: Identify the Semi-Major and Semi-Minor Axes ($a$ and $b$). From the standard form $\frac{x^2}{144} + \frac{y^2}{25} = 1$, we can identify the denominators under $x^2$ and $y^2$:

• The denominator under $x^2$ is $144$.
• The denominator under $y^2$ is $25$.