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 $Ax+By+C=0$ is $m=-\frac{A}{B}$
It is known that parallel lines have the same slope
$\therefore$ slope of other line $=m=-\frac{A}{B}$
Thus the equation of the line passing through point$(x_1,y_1)$ and having a slope $m=-\frac{A}{B}$ is,
$y-y_1=m(x-x_1)$
$\Rightarrow y-y_1=-\frac{A}{B}(x-x_1)$
$\Rightarrow B(y-y_1)=A(x-x_1)$
$\Rightarrow A(x-x_1)+B(y-y_1)=0$
Hence, the line through point $(x_1,y_1)$ and parallel to line $Ax+By+C=0$ is $A(x-x_1)+B(y-y_1)=0$