A mirror is placed in x-y plane at an angle of 30° to the x axis . An object is placed at the coordinate (2,0). What is the coordinate of the image?

Question

Answer

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,

Answer

(1,sqrt(3)) or (1,-sqrt(3)) as the mirror can be kept in two ways such that it's angle with x-axis ( default to be taken as +ve side) is 30 degrees.

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