A ray of light passing through the point reflects on the -axis at point A and the reflected ray passes through the point . Find the coordinates of A.
The coordinates of point A are (13/5, 0).
Explanation
This problem uses the concept of reflection. When a ray reflects off a surface (here, the x-axis), the angle of incidence equals the angle of reflection. A key insight from the textbook (Example 13 context) is that straight lines work as plane mirrors. The image of the incident point in the mirror line lies on the path of the reflected ray. Since the x-axis is y=0, the image of point (1,2) in the x-axis is (1,-2). The reflected ray passes through this image point and the given point (5,3), so point A lies at the intersection of this line with the x-axis.
Solution Steps
Step 1: Find the image of point P(1, 2) in the x-axis (y = 0). Since the x-axis acts as a mirror, the image of (1, 2) is P'(1, -2).
Step 2: The reflected ray appears to come from the image point P'(1, -2). Therefore, point A lies on the line joining P'(1, -2) and Q(5, 3).
Step 3: Find the equation of line passing through P'(1, -2) and Q(5, 3). Using the slope formula from context: m = (y₂ - y₁)/(x₂ - x₁) = (3 - (-2))/(5 - 1) = 5/4.
Step 4: Using point-slope form from context: y - y₁ = m(x - x₁), we get y - (-2) = (5/4)(x - 1), or y + 2 = (5/4)(x - 1).
Step 5: Point A lies on the x-axis, so y = 0. Substituting: 0 + 2 = (5/4)(x - 1), giving x - 1 = 8/5, so x = 13/5.
Step 6: Therefore, the coordinates of A are (13/5, 0).