Question

select the correct answer from each drop - down menu.
a ray of light is reflected from a mirror such that the reflected ray is perpendicular to the original ray, as shown in the diagram. the equation of the reflected ray is
the point
does not lie on the reflected ray.

Explanation:

Part 1: Equation of the Reflected Ray

Step 1: Find the slope of the original ray

The original ray has the equation \( y - x = 4 \), which can be rewritten as \( y = x + 4 \). The slope (\( m_1 \)) of this line is \( 1 \) (since it's in the form \( y = mx + b \), where \( m = 1 \)).

Step 2: Determine the slope of the reflected ray

If two lines are perpendicular, the product of their slopes is \( -1 \). Let the slope of the reflected ray be \( m_2 \). So, \( m_1 \times m_2 = -1 \). Substituting \( m_1 = 1 \), we get \( 1 \times m_2 = -1 \), so \( m_2 = -1 \).

Step 3: Find the equation of the reflected ray

The reflected ray passes through the point \( C(1, 5) \). Using the point - slope form of a line \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(1,5) \) and \( m=-1 \).
Substitute the values: \( y - 5=-1(x - 1) \)
Simplify: \( y - 5=-x + 1 \)
Rearrange to get the standard form: \( y+x=6 \)

Part 2: Point not on the reflected ray

We will check each point (assuming the possible points are related to the lines, but since the second drop - down is not fully visible, we can use the equation \( y + x=6 \) to check. Let's assume the points are related to the given lines. For a point \( (x,y) \) to lie on the line \( y + x=6 \), \( x + y \) should equal \( 6 \).

For example, if we take a point like (2,4): \( 2 + 4=6 \), so it lies on the line. If we take a point like (3,4): \( 3+4 = 7
eq6 \), so it does not lie on the line. But since the second drop - down is not provided with options, we can focus on the first part.

Answer:

(for the first drop - down):
The equation of the reflected ray is \( y + x = 6 \)