Find the image of the point with respect to the line assuming the line to be a plane mirror.
The image of the point (3, 8) with respect to the line x + 3y = 7 is (-1, -4).
Explanation
This question follows the exact pattern of Example 13 from the textbook. The key concept is that when a line acts as a plane mirror, it becomes the perpendicular bisector of the line segment joining the point and its image. Two conditions must be satisfied: (1) the line joining the point and its image must be perpendicular to the mirror line, and (2) the midpoint of this segment must lie on the mirror line. These two conditions give us two equations that can be solved simultaneously to find the coordinates of the image point.
Solution Steps
Step 1: Let Q(h, k) be the image of point P(3, 8) in the line x + 3y = 7. The line acts as the perpendicular bisector of segment PQ.
Step 2: Find the slope of the given line: x + 3y = 7 can be written as y = -x/3 + 7/3, so slope = -1/3.
Step 3: Apply perpendicularity condition: Slope of PQ × Slope of line = -1. So, (k-8)/(h-3) × (-1/3) = -1, which gives (k-8)/(h-3) = 3, or k - 3h = -1 ... (1)
Step 4: Apply midpoint condition: The midpoint of PQ is ((h+3)/2, (k+8)/2). This must satisfy x + 3y = 7. So, (h+3)/2 + 3(k+8)/2 = 7, which gives h + 3k = -13 ... (2)
Step 5: Solve equations (1) and (2) simultaneously. From (1): k = 3h - 1. Substituting in (2): h + 3(3h - 1) = -13, giving h = -1. Then k = 3(-1) - 1 = -4.
Step 6: Therefore, the image of point (3, 8) is (-1, -4).