A mirror is placed in x-y plane at an angle of 30° to the x axis . An ...
Approach:
- Find the angle of incidence
- Use the formula to find the angle of reflection
- Find the slope of the line
- Find the equation of the line
- Find the intersection point of the line and the mirror
- Find the equation of the perpendicular bisector
- Find the intersection point of the perpendicular bisector and the mirror
Solution:
Step 1: Finding the angle of incidence
The angle of incidence is the angle between the mirror and the line connecting the object and the mirror.
Let's draw a diagram:
As we can see from the diagram, the angle of incidence is 60° (since the mirror is placed at an angle of 30° to the x-axis).
Step 2: Finding the angle of reflection
The angle of reflection is equal to the angle of incidence.
Therefore, the angle of reflection is also 60°.
Step 3: Finding the slope of the line
The slope of the line connecting the object and the mirror can be found using the formula:
m = tan(θ), where θ = the angle of reflection
Therefore, m = tan(60°) = √3
Step 4: Finding the equation of the line
The equation of the line can be found using the point-slope formula:
y - y1 = m(x - x1), where (x1, y1) = (2, 0)
Therefore, the equation of the line is y = √3(x - 2)
Step 5: Finding the intersection point of the line and the mirror
Since the mirror is at an angle of 30° to the x-axis, its equation is y = (x/√3).
Therefore, we can find the intersection point by solving the system of equations:
y = √3(x - 2)
y = x/√3
Solving for x and y, we get:
x = 2/3, y = 2√3/3
Step 6: Finding the equation of the perpendicular bisector
The perpendicular bisector of the line joining the object and its image is the line passing through the midpoint of the line and perpendicular to it.
The midpoint of the line is (2 + 2/3)/2, (0 + 2√3/3)/2 = (7/3, √3/3)
Since the slope of the line is √3,