This problem is a sophisticated blend of coordinate geometry principles, particularly reflection of light, and the fundamental properties of an ellipse. Our goal is to determine the equation of the "other directrix" of an ellipse, given information about one directrix (derived from the reflected ray), the eccentricity, and a specific distance related to its focus. We will strategically break down the problem into manageable steps, explaining the rationale behind each.

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1. Problem Overview and Strategic Approach

We are tasked with finding the equation of the second directrix ($D_2$) of an ellipse. We are provided with:

1. Information about a ray of light that defines the first directrix ($D_1$).
2. The eccentricity ($e$) of the ellipse.
3. The distance of the nearer focus from $D_1$.

Our strategy will be as follows:

• Step 1: Determine the equation of the first directrix ($D_1$). This involves using the principle of reflection for the given ray of light.
• Step 2: Utilize the ellipse's eccentricity and the given focus-directrix distance to find the semi-major axis ($a$). This will establish a key parameter of the ellipse.
• Step 3: Calculate the total distance between the two directrices. This distance is directly related to $a$ and $e$.
• Step 4: Use the equation of $D_1$ and the calculated inter-directrix distance to find the possible equations for $D_2$. Since directrices are parallel, this step involves finding the constant term of the second directrix's equation.

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2. Determining the Equation of the First Directrix ($D_1$) - The Reflected Ray

Key Concept: Principle of Reflection using Image Points For a ray of light originating from point $A$, reflecting off a line $L$, and passing through point $B$, the reflected path can be found by connecting the image of $A$ (say, $A'$) with respect to line $L$ to point $B$. The line segment $A'B$ represents the path of the reflected ray. This method simplifies reflection problems into finding the equation of a straight line.

Why this step is taken: The problem states that the reflected ray is the directrix of the ellipse. Therefore, our first objective is to find the algebraic equation of this line. Using the image point method is the most efficient way to achieve this.

• Given Information:

  • Source point $A = (2, 1)$.
  • Reflection surface: y-axis, which has the equation $x=0$.
  • Point on the reflected ray $B = (5, 3)$.

• Step-by-Step Calculation:

  1. Find the image of point $A(2, 1)$ across the y-axis ($x=0$). When a point $(x, y)$ is reflected across the y-axis, its x-coordinate changes sign, while its y-coordinate remains the same. The image point $A'$ will be $(-x, y)$. Therefore, the image of $A(2, 1)$ is $A'(-2, 1)$.

  2. Determine the equation of the line passing through $A'(-2, 1)$ and $B(5, 3)$. This line represents the reflected ray, which is our directrix $D_1$. First, calculate the slope ($m$) of this line: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{5 - (-2)} = \frac{2}{5 + 2} = \frac{2}{7}$ Now, use the point-slope form of the line equation, $y - y_1 = m(x - x_1)$. We can use either $A'$ or $B$. Let's use $A'(-2, 1)$: $y - 1 = \frac{2}{7}(x - (-2))$ $y - 1 = \frac{2}{7}(x + 2)$ To eliminate the fraction and get the standard form $Ax + By + C = 0$, multiply both sides by 7: $7(y - 1) = 2(x + 2)$ $7y - 7 = 2x + 4$ Rearrange the terms: $2x - 7y + 4 + 7 = 0$ $2x - 7y + 11 = 0$ This is the equation of our first directrix, $D_1$. $\mathbf{D_1: 2x - 7y + 11 = 0}$

Tip for Reflection Problems: Always ensure you are using the image point for the reflected ray. A common mistake is to accidentally use the original source point, which would lead to an incorrect line equation.

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3. Using Ellipse Properties to Find the Semi-Major Axis ($a$)

Key Concept: Distance from a Focus to its Directrix For an ellipse with semi-major axis $a$ and eccentricity $e$:

• The distance from the center to a focus is $ae$.
• The distance from the center to a directrix is $a/e$.
• The distance from a focus to its corresponding directrix (the one closer to it) is $\frac{a}{e} - ae$. This is often referred to as the distance of the "nearer focus" from the directrix.

Why this step is taken: We are given the eccentricity ($e$) and the distance from the nearer focus to the directrix $D_1$. This specific relationship allows us to set up an equation to solve for the semi-major axis $a$, which is a fundamental parameter needed to determine the overall dimensions of the ellipse and the distance between its directrices.

• Given Information:

  • Eccentricity $e = \frac{1}{3}$.
  • Distance of the nearer focus from the directrix $D_1$ is $\frac{8}{\sqrt{53}}$.

• Step-by-Step Calculation:

  1. Set up the equation using the formula for the distance of the nearer focus from its directrix: $\frac{a}{e} - ae = \text{Given Distance}$ $\frac{a}{e} - ae = \frac{8}{\sqrt{53}}$
  2. Substitute the value of $e = \frac{1}{3}$ into the equation: $\frac{a}{(1/3)} - a\left(\frac{1}{3}\right) = \frac{8}{\sqrt{53}}$ $3a - \frac{a}{3} = \frac{8}{\sqrt{53}}$
  3. Combine the terms involving $a$: Find a common denominator: $\frac{9a - a}{3} = \frac{8}{\sqrt{53}}$ $\frac{8a}{3} = \frac{8}{\sqrt{53}}$
  4. Solve for $a$: Multiply both sides by $\frac{3}{8}$: $a = \frac{8}{\sqrt{53}} \times \frac{3}{8}$ $a = \frac{3}{\sqrt{53}}$ So, the semi-major axis of the ellipse is $a = \frac{3}{\sqrt{53}}$.

Common Mistake: Confusing the distance from a focus to its directrix ($a/e - ae$ or $a/e + ae$) with the distance between the two directrices ($2a/e$). The problem explicitly states "distance of the nearer focus from this directrix," which corresponds to $\frac{a}{e} - ae$.

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4. Calculating the Distance Between the Two Directrices

Key Concept: Distance Between Directrices For an ellipse, the two directrices are parallel lines symmetrically positioned with respect to the center. The total distance between them is given by the formula $\frac{2a}{e}$.

Why this step is taken: We have the equation of one directrix ($D_1$) and we need to find the equation of the second directrix ($D_2$). Since directrices are always parallel, $D_2$ will have the same slope as $D_1$. To precisely locate $D_2$, we need to know how far it is from $D_1$. The distance between the directrices provides this crucial information.

• Step-by-Step Calculation:
  1. Use the formula for the distance between the two directrices: $\text{Distance between directrices} = \frac{2a}{e}$
  2. Substitute the values of $a = \frac{3}{\sqrt{53}}$ (calculated in Step 3) and $e = \frac{1}{3}$ (given): $\text{Distance} = \frac{2 \left(\frac{3}{\sqrt{53}}\right)}{\frac{1}{3}}$ $\text{Distance} = 2 \times \frac{3}{\sqrt{53}} \times 3 \quad \text{(multiplying by the reciprocal of } \frac{1}{3})$ $\text{Distance} = \frac{18}{\sqrt{53}}$ So, the distance between the two directrices is $\frac{18}{\sqrt{53}}$.

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5. Finding the Equation of the Other Directrix ($D_2$)

Key Concept: Distance Between Parallel Lines The distance between two parallel lines given by $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is calculated using the formula: $\text{Distance} = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$

Why this step is taken: We have the equation of the first directrix $D_1: 2x - 7y + 11 = 0$. We know the second directrix $D_2$ must be parallel to $D_1$, so its equation will be of the form $2x - 7y + C = 0$ for some constant $C$. We also know the exact distance between $D_1$ and $D_2$ from the previous step. This information allows us to set up an equation using the parallel line distance formula to solve for the unknown constant $C$.

• Given Information:

  • Equation of $D_1: 2x - 7y + 11 = 0$. (Here, $A=2$, $B=-7$, $C_1=11$).
  • Distance between directrices $= \frac{18}{\sqrt{53}}$ (calculated in Step 4).

• Step-by-Step Calculation:

  1. Let the equation of the other directrix be $D_2: 2x - 7y + C = 0$.
  2. Apply the distance formula for parallel lines: $\frac{|C_1 - C|}{\sqrt{A^2 + B^2}} = \text{Distance}$ $\frac{|11 - C|}{\sqrt{2^2 + (-7)^2}} = \frac{18}{\sqrt{53}}$
  3. Calculate the denominator $\sqrt{A^2 + B^2}$: $\sqrt{2^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53}$
  4. Substitute this back into the distance equation: $\frac{|11 - C|}{\sqrt{53}} = \frac{18}{\sqrt{53}}$
  5. Solve for $C$: Multiply both sides by $\sqrt{53}$: $|11 - C| = 18$ The absolute value equation gives two possibilities:

     • Possibility 1: $11 - C = 18$ $-C = 18 - 11$ $-C = 7$ $C = -7$ So, one possible equation for $D_2$ is $2x - 7y - 7 = 0$.

     • Possibility 2: $11 - C = -18$ $-C = -18 - 11$ $-C = -29$ $C = 29$ So, the other possible equation for $D_2$ is $2x - 7y + 29 = 0$.

  Therefore, the equations of the other directrix can be $2x - 7y - 7 = 0$ or $2x - 7y + 29 = 0$.

Important Note: The two solutions for $C$ arise because the second directrix can be on either side of the first directrix, at the specified distance. The orientation of the ellipse relative to $D_1$ determines which solution is the "actual" $D_2$, but without more information (like the center or foci), both are mathematically valid possibilities for the other directrix.

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6. Comparing with Options and Final Answer

We have found the possible equations for the other directrix to be:

1. $2x - 7y - 7 = 0$
2. $2x - 7