A ray of light is sent along the line x-6y=8, after refracting along the line x+y=1,it enters the opposite side after turning by 15° away from the line x+y=1.Find the equation of the line along which refracted ray travels.

To find the equation of the line along which the refracted ray travels, we need to follow a series of logical steps involving the given lines and the angle of refraction. Let's break this down step by step.

Step 1: Understanding the Given Lines

We have two lines defined by their equations:

• The first line is given by x - 6y = 8.
• The second line is x + y = 1.

Step 2: Finding the Direction of the Incident Ray

To determine the direction of the incident ray, we can rewrite the first line in slope-intercept form (y = mx + b). Rearranging x - 6y = 8 gives us:

6y = x - 8 or y = (1/6)x - 4/3.

From this, we can see that the slope (m) of the incident ray is 1/6.

Step 3: Finding the Angle of the Incident Ray

The angle of the incident ray with respect to the positive x-axis can be calculated using the arctangent of the slope:

θ₁ = arctan(1/6).

Step 4: Finding the Direction of the Refracted Ray

The second line, where the ray refracts, is x + y = 1. We can also express this in slope-intercept form:

y = -x + 1.

Here, the slope is -1. The angle of this line with respect to the positive x-axis is:

θ₂ = arctan(-1) = -45°.

Step 5: Calculating the Angle of Refraction

After refracting along the line x + y = 1, the ray turns by 15° away from this line. Therefore, the angle of the refracted ray (θ₃) can be calculated as:

θ₃ = θ₂ + 15° = -45° + 15° = -30°.

Step 6: Finding the Slope of the Refracted Ray

The slope of the refracted ray can be determined from the angle θ₃:

m₃ = tan(-30°) = -√3/3.

Step 7: Writing the Equation of the Refracted Ray

Now that we have the slope of the refracted ray, we can use the point of refraction to find its equation. The point of refraction is where the two lines intersect. To find this point, we can solve the equations:

• x - 6y = 8
• x + y = 1

Finding the Intersection Point

From the second equation, we can express x in terms of y:

x = 1 - y.

Substituting this into the first equation:

(1 - y) - 6y = 8

1 - 7y = 8

-7y = 7 or y = -1.

Substituting back to find x:

x = 1 - (-1) = 2.

Thus, the point of intersection is (2, -1).

Step 8: Final Equation of the Refracted Ray

Using the point-slope form of the line equation, we can write:

y - y₁ = m(x - x₁), where (x₁, y₁) is (2, -1) and m is -√3/3.

Substituting these values gives:

y + 1 = -√3/3(x - 2).

Rearranging this, we find:

y = -√3/3x + 2√3/3 - 1.

Final Result

The equation of the line along which the refracted ray travels is:

y = -√3/3x + (2√3/3 - 1).

This gives us the complete description of the path of the refracted ray after it turns away from the line x + y = 1.