A ray of light through (2,1) is reflected at a point P on the y-axis and then passes through the point (5,3). If this reflected ray is the directrix of an ellipse with eccentricity `(1)/(3)` and the distance of the nearer focus from this directrix is `(8)/(sqrt53)`, then the equation of the other directrix can be

Options

A. `11x+7y+8=0 or 11x+7y-15=0`
B. `11x-7y-8=0 or 11x+7y+15=0`
C. `2x-7y+29=0 or 2x-7y-7=0`
D. `2x-7y-39=0 or 2x-7y-7=0`

Answer

To solve the given problem step by step, we will follow the instructions provided in the video transcript while ensuring clarity in each step.

### Step 1: Understand the Geometry of the Problem
A ray of light passes through the point (2, 1) and reflects at a point P on the y-axis, then passes through the point (5, 3). We need to find the equation of the other directrix of an ellipse, given that the reflected ray serves as the directrix.

### Step 2: Determine the Slope of the Incoming Ray
The slope of the ray from (2, 1) to (5, 3) can be calculated as follows:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{5 - 2} = \frac{2}{3} \]

### Step 3: Write the Equation of the Incoming Ray
Using the point-slope form of the line equation:
\[ y - 1 = \frac{2}{3}(x - 2) \]
Simplifying this:
\[ y - 1 = \frac{2}{3}x - \frac{4}{3} \implies y = \frac{2}{3}x + \frac{1}{3} \]

### Step 4: Determine the Point of Reflection on the y-axis
Let the point of reflection P be (0, y). The slope of the line from (2, 1) to (0, y) is:
\[ \text{slope} = \frac{y - 1}{0 - 2} = \frac{1 - y}{2} \]
By the law of reflection, the angle of incidence equals the angle of reflection. Therefore, we can set the slopes equal:
\[ \frac{1 - y}{2} = -\frac{2}{3} \]
Cross-multiplying gives:
\[ 3(1 - y) = -4 \implies 3 - 3y = -4 \implies 3y = 7 \implies y = \frac{7}{3} \]
Thus, the point of reflection P is (0, \( \frac{7}{3} \)).

### Step 5: Write the Equation of the Reflected Ray
The reflected ray passes through (0, \( \frac{7}{3} \)) and (5, 3). The slope of the reflected ray is:
\[ \text{slope} = \frac{3 - \frac{7}{3}}{5 - 0} = \frac{\frac{9 - 7}{3}}{5} = \frac{2/3}{5} = \frac{2}{15} \]
Using the point-slope form again:
\[ y - \frac{7}{3} = \frac{2}{15}(x - 0) \]
Simplifying gives:
\[ y = \frac{2}{15}x + \frac{7}{3} \]

### Step 6: Convert to Standard Form
To express this in standard form:
\[ 15y = 2x + 35 \implies 2x - 15y + 35 = 0 \]

### Step 7: Determine the Directrix of the Ellipse
The reflected ray serves as the directrix of an ellipse with eccentricity \( e = \frac{1}{3} \) and the distance of the nearer focus from this directrix is given as \( \frac{8}{\sqrt{53}} \). The distance from the focus to the directrix is given by \( \frac{a}{e} \) where \( a \) is the semi-major axis.

### Step 8: Calculate the Semi-Major Axis
From the relationship:
\[ \frac{a}{e} = \frac{8}{\sqrt{53}} \implies a = e \cdot \frac{8}{\sqrt{53}} = \frac{1}{3} \cdot \frac{8}{\sqrt{53}} = \frac{8}{3\sqrt{53}} \]

### Step 9: Calculate the Distance Between Directrices
The distance between the two directrices is given by:
\[ \frac{2a}{e} = \frac{2 \cdot \frac{8}{3\sqrt{53}}}{\frac{1}{3}} = \frac{16}{\sqrt{53}} \]

### Step 10: Find the Other Directrix
Let the equation of the other directrix be \( 2x - 15y + \lambda = 0 \). The distance from the point (0, \( \frac{7}{3} \)) to this directrix must equal \( \frac{16}{\sqrt{53}} \):
\[ \frac{|0 - 15 \cdot \frac{7}{3} + \lambda|}{\sqrt{2^2 + (-15)^2}} = \frac{16}{\sqrt{53}} \]
Calculating the left side:
\[ \sqrt{4 + 225} = \sqrt{229} \]
Thus:
\[ \frac{|\lambda - 35|}{\sqrt{229}} = \frac{16}{\sqrt{53}} \]
Cross-multiplying gives:
\[ |\lambda - 35| = \frac{16 \sqrt{229}}{\sqrt{53}} \]

### Step 11: Solve for λ
This results in two equations:
1. \( \lambda - 35 = \frac{16 \sqrt{229}}{\sqrt{53}} \)
2. \( \lambda - 35 = -\frac{16 \sqrt{229}}{\sqrt{53}} \)

Solving these will yield the values for \( \lambda \).

### Final Step: Write the Equations of the Other Directrix
Substituting the values of \( \lambda \) back into the equation \( 2x - 15y + \lambda = 0 \) will give the equations of the other directrix.