This solution provides a detailed, step-by-step approach to solve the problem, integrating concepts from coordinate geometry and conic sections. Each step is accompanied by a clear explanation of the underlying logic and relevant formulas, along with tips to avoid common pitfalls.

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1. Key Concepts and Formulas Used

Before diving into the solution, let's review the essential mathematical tools we'll be employing:

• Intercepts of a Line:

  • To find the x-intercept (where the line crosses the x-axis), set $y=0$ in the line equation and solve for $x$. The point will be $(\text{x-intercept}, 0)$.
  • To find the y-intercept (where the line crosses the y-axis), set $x=0$ in the line equation and solve for $y$. The point will be $(0, \text{y-intercept})$.

• Section Formula (Internal Division):

  • If a point $P(x, y)$ divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m:n$, its coordinates are given by: $P\left(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}\right)$

• Standard Equation of an Ellipse:

  • For an ellipse centered at the origin with its major axis along the x-axis, the equation is: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ where $a$ is the length of the semi-major axis, $b$ is the length of the semi-minor axis, and $a > b > 0$.

• Properties of an Ellipse (Major Axis along x-axis):

  • Foci: The foci are located at $S(\pm ae, 0)$, where $e$ is the eccentricity.
  • Directrices: The equations of the directrices are $x = \pm \frac{a}{e}$. Each focus corresponds to a specific directrix (e.g., focus $(ae, 0)$ corresponds to directrix $x = \frac{a}{e}$).
  • Eccentricity ($e$): It's a measure of how "stretched" the ellipse is. It's related to $a$ and $b$ by the equation: $e = \sqrt{1 - \frac{b^2}{a^2}}$ This can be rearranged to find $b^2$: $b^2 = a^2(1-e^2)$ For an ellipse, $0 < e < 1$.

• Length of Latus Rectum:

  • For an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, the length of the latus rectum (a chord passing through a focus and perpendicular to the major axis) is: $\text{Length of Latus Rectum} = \frac{2b^2}{a}$

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2. Step 1: Determine the Coordinates of Points A and B

The problem states that points $A(\alpha, 0)$ and $B(0, \beta)$ lie on the line $5x + 7y = 50$. These points are, by definition, the x-intercept and y-intercept of the line, respectively. Our first step is to find their exact coordinates.

• Finding Point A ($\alpha, 0$): Point $A$ lies on the x-axis, which means its y-coordinate is $0$.

  • Why: We substitute $y=0$ into the line equation to find the x-coordinate ($\alpha$) where the line intersects the x-axis. $5\alpha + 7(0) = 50$ $5\alpha = 50$ $\alpha = 10$ So, point $A$ is $(10, 0)$.

• Finding Point B ($0, \beta$): Point $B$ lies on the y-axis, which means its x-coordinate is $0$.

  • Why: We substitute $x=0$ into the line equation to find the y-coordinate ($\beta$) where the line intersects the y-axis. $5(0) + 7\beta = 50$ $7\beta = 50$ $\beta = \frac{50}{7}$ So, point $B$ is $\left(0, \frac{50}{7}\right)$.

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3. Step 2: Find the Coordinates of Point P

The problem states that point $P$ divides the line segment $AB$ internally in the ratio $7:3$. We will use the section formula for internal division.

• Why: The coordinates of point P are crucial because its x-coordinate will directly relate to the focus of the ellipse, as described in the problem statement.

• Let $A(x_1, y_1) = (10, 0)$ and $B(x_2, y_2) = \left(0, \frac{50}{7}\right)$.

• The ratio is $m:n = 7:3$.

• Applying the Section Formula: $P\left(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}\right)$ Substitute the values: $P\left(\frac{7(0) + 3(10)}{7+3}, \frac{7\left(\frac{50}{7}\right) + 3(0)}{7+3}\right)$ $P\left(\frac{0 + 30}{10}, \frac{50 + 0}{10}\right)$ $P\left(\frac{30}{10}, \frac{50}{10}\right)$ $P(3, 5)$ Thus, the coordinates of point $P$ are $(3, 5)$.

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4. Step 3: Relate the Ellipse Properties to the Given Information

The ellipse $E$ is given by $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. This is a standard form for an ellipse centered at the origin with its major axis along the x-axis (since $a>b$ is implicitly assumed for this standard form). We need to use the given directrix and the information about point $P$ and focus $S$ to find relationships for $a$ and $e$.

• Using the Directrix Information: The given directrix is $3x - 25 = 0$, which can be rewritten as $x = \frac{25}{3}$.

  • Why: For an ellipse with its major axis along the x-axis, the equations of the directrices are $x = \pm \frac{a}{e}$. Since the given directrix $x = \frac{25}{3}$ has a positive x-value, it corresponds to the focus with a positive x-coordinate, $S(ae, 0)$. Therefore, we have our first equation: $\frac{a}{e} = \frac{25}{3} \quad \text{(Equation 1)}$
  • Tip: Always ensure you match the sign of the directrix with the corresponding focus. A positive directrix ($x=a/e$) corresponds to the focus $S(ae, 0)$, and a negative directrix ($x=-a/e$) corresponds to $S'(-ae, 0)$.

• Using the Focus and Point P Information: The problem states that the corresponding focus is $S$. Based on the directrix $x = \frac{a}{e}$, the corresponding focus is $S(ae, 0)$. We are told that "from $S$, the perpendicular on the x-axis passes through $P$".

  • Why: Since the focus $S(ae, 0)$ already lies on the x-axis, the "perpendicular on the x-axis from S" is essentially the vertical line passing through $S$, which has the equation $x = ae$.
  • If this vertical line $x = ae$ passes through point $P(3, 5)$, it means that the x-coordinate of $P$ must be equal to $ae$. Therefore, we have our second equation: $ae = 3 \quad \text{(Equation 2)}$
  • Common Misinterpretation: A common mistake here is to assume that $P$ is the focus, or that the y-coordinate of $S$ is 5. However, $S$ is a focus on the x-axis, so its y-coordinate is 0. The statement means the vertical line $x=ae$ passes through $P(3,5)$, which implies $x_P = ae$.

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5. Step 4: Calculate the Values of 'a', 'e', and 'b'

Now we have a system of two equations with two unknowns ($a$ and $e$):

1. $\frac{a}{e} = \frac{25}{3}$
2. $ae = 3$

• Why: We need to find $a$ and $b^2$ to calculate the length of the latus rectum, which is our ultimate goal.

• Finding 'a': To eliminate $e$ and solve for $a$, we can multiply Equation 1 by Equation 2: $\left(\frac{a}{e}\right) \times (ae) = \frac{25}{3} \times 3$ $a^2 = 25$ $a = 5 \quad (\text{Since } a \text{ represents a semi-axis length, it must be positive})$

• Finding 'e': Substitute the value of $a=5$ into Equation 2: $5e = 3$ $e = \frac{3}{5}$

  • Check: For an ellipse, the eccentricity $e$ must be between 0 and 1 ($0 < e < 1$). Our calculated value $e = 3/5$ satisfies this condition, confirming our calculations are on track.

• Finding 'b' (or $b^2$): We use the fundamental relationship for an ellipse: $b^2 = a^2(1-e^2)$.

  • Why: This formula directly connects $a$, $e$, and $b^2$, allowing us to find $b^2$ which is required for the latus rectum formula. $b^2 = (5)^2 \left(1 - \left(\frac{3}{5}\right)^2\right)$ $b^2 = 25 \left(1 - \frac{9}{25}\right)$ $b^2 = 25 \left(\frac{25 - 9}{25}\right)$ $b^2 = 25 \left(\frac{16}{25}\right)$ $b^2 = 16$

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6. Step 5: Calculate the Length of the Latus Rectum

Now that we have $a=5$ and $b^2=16$, we can calculate the length of the latus rectum.

• Why: This is the final quantity requested by the problem.
• The formula for the length of the latus rectum for an ellipse with its major axis along the x-axis is $\frac{2b^2}{a}$. $\text{Length of Latus Rectum} = \frac{2b^2}{a}$ Substitute the values: $\text{Length of Latus Rectum} = \frac{2(16)}{5}$ $\text{Length of Latus Rectum} = \frac{32}{5}$

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7. Conclusion and Key Takeaways

The length of the latus rectum of ellipse $E$ is $\frac{32}{5}$.

Comparing this with the given options: (A) $\frac{25}{3}$ (B) $\frac{25}{9}$ (C) $\frac{32}{5}$ (D) $\frac{32}{9}$

The correct answer is (C).

This problem effectively tests your understanding of various coordinate geometry and conic section concepts. Key takeaways include:

• Accurate calculation of line intercepts and application of the section formula.
• Careful interpretation of the properties of an ellipse, especially the relationship between foci, directrices, eccentricity, and the definition of a "perpendicular from a focus to the x-axis passing through a point."