Question

Prove that the line through the point $(x_1, y_1)$ and parallel to the line $Ax + By + C = 0$ is $A (x -x_1) + B (y - y_1) = 0.$

Answer 1

The slope of line $Ax + By + C = 0$ or $y =\frac{ -A}{B}x –\frac{ C}{B}$ is $m = \frac{-A}{B}$
It is known that parallel lines have the same slope.

∴Slope of the other line $=m = \frac{-A}{B}$
The equation of the line passing through point $(x_1, y_1)$ and having a slope $m = \frac{-A}{B}$ is

$y – y_1 = m (x – x_1)$

$y – y_1= \frac{-A}{B }(x – x_1)$

$B (y – y_1) = -A (x – x_1)$

$∴ A(x – x_1) + B(y – y_1) = 0$

Hence, the line through point $(x_1, y_1)$ and parallel to line $Ax + By + C = 0$ is $A (x - x_1) + B (y \- y_1) = 0$