Question

Write the coordinates of the image of the point (3, 8) in the line x + 3y − 7 = 0.

Answer 1

Let the given point be A(3,8) and its image in the line x + 3y − 7 = 0 is B(h,k).
The midpoint of AB is \[\left( \frac{3 + h}{2}, \frac{8 + k}{2} \right)\] that lies on the line x + 3y − 7 = 0. 

\[\therefore \frac{3 + h}{2} + 3 \times \frac{8 + k}{2} - 7 = 0\]

\[h + 3k + 13 = 0\]         ... (1)

AB and the line x + 3y − 7 = 0 are perpendicular.

\[\therefore\text {  Slope of AB } \times \text { Slope of the line } = - 1\]

\[ \Rightarrow \frac{k - 8}{h - 3} \times \frac{- 1}{3} = - 1\]

\[\Rightarrow 3h - k - 1 = 0\]       ... (2)
Solving (1) and (2), we get:
(h, k) = (−1, −4)
Hence, the image of the point (3,8) in the line x + 3y − 7 = 0 is (−1,−4).