a ray of light travelling along the line op is reflected by a line mirror 2 x- 3 y+1=0 the point of incidence being p(1,1) the reflected ray travel along pq is a given reflected by mirror 2x - 3y - 1 = 0 the point of incidence being Q from Q, ray move along QR, where R lies on the line 2x - 3y + 1 = 0 The ordinate of point R is

To find the ordinate of the point \( R \), we need to consider the reflection of the rays at each mirror. 

### Step 1: Analyze the given information
- The line OP represents the path of the initial ray.
- The first mirror is given by \( 2x - 3y + 1 = 0 \).
- The point of incidence \( P(1, 1) \) is on the first mirror.
- The second mirror is given by \( 2x - 3y - 1 = 0 \).
- The point of incidence on the second mirror is \( Q \).
- The ray is finally reflected again to a point \( R \) that lies on the first mirror \( 2x - 3y + 1 = 0 \).

### Step 2: Understand the reflection property
When a ray of light reflects off a mirror, the angle of incidence equals the angle of reflection. The slopes of the incident and reflected rays can be related using this property.

### Step 3: Equation of the line OP and reflection at P
The ray OP is reflected at \( P(1, 1) \) off the mirror \( 2x - 3y + 1 = 0 \). Let's assume the equation of OP is \( y = mx + c \).

Since \( P(1, 1) \) lies on the line \( y = mx + c \):
\[
1 = m(1) + c \implies c = 1 - m
\]
So, the equation of line OP is:
\[
y = mx + (1 - m)
\]

### Step 4: Find the slope of the reflected ray PQ
The slope of the mirror \( 2x - 3y + 1 = 0 \) is \( \frac{2}{3} \). The slope of the reflected ray is determined by applying the reflection formula:
\[
m_{\text{reflected}} = \frac{m_1m_2 - 1}{m_2^2 - m_1m_2}
\]
where \( m_1 \) is the slope of the incident ray, and \( m_2 \) is the slope of the mirror. 

After reflection, the ray travels along PQ and strikes the second mirror at point \( Q \).

### Step 5: Find the coordinates of \( R \)
\( R \) lies on the line \( 2x - 3y + 1 = 0 \). The reflected ray QR reflects again at the second mirror \( 2x - 3y - 1 = 0 \) and returns to the first mirror. 

The points \( P(1, 1) \) and \( Q \) have the same x-coordinate, so the ray QR will be vertical.

Given the symmetry and the fact that \( R \) lies on the line \( 2x - 3y + 1 = 0 \), we substitute \( x = 1 \) in the equation \( 2x - 3y + 1 = 0 \):
\[
2(1) - 3y + 1 = 0 \implies 2 - 3y + 1 = 0 \implies 3 - 3y = 0 \implies y = 1
\]

### Conclusion
The ordinate of point \( R \) is \( y = 1 \).

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Would you like details or have any questions? Here are 8 related questions for further practice:

1. How do you determine the slope of a line given its equation?
2. What is the formula for the reflection of a line across another line?
3. How do you find the point of incidence when a ray reflects off a mirror?
4. What is the relationship between the incident and reflected angles in a reflection problem?
5. How do you find the equation of a line given a point and a slope?
6. Can you describe how to find the coordinates of the reflection of a point across a line?
7. What is the significance of the slopes in determining the path of reflected rays?
8. How does the symmetry of the problem influence the final result in reflection scenarios?

**Tip:** When working with reflections, always double-check the relationship between the incident and reflected angles to ensure accuracy in the direction of the reflected ray.

Explore the reflection of light problem where a ray travels along OP, reflects at mirror 2x - 3y + 1 = 0 at point P(1,1), then along PQ, reflecting again at mirror 2x - 3y - 1 = 0 at point Q, finding point R on the line 2x - 3y + 1 = 0. Understand reflection formulas and equations involved. Suitable for high school students.

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