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.

$$\begin{gathered} \therefore \mathrm{Slope}\mathrm{of}AB\times \mathrm{Slope}\mathrm{of}\mathrm{the}\mathrm{line}=-1 \\ \Rightarrow \frac{k-8}{h-3}\times \frac{-1}{3}=-1 \end{gathered}$$

$\Rightarrow 3h-k-1=0$ ... (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).