Answer to Question #101468 in Analytic Geometry for Engr. Ibeto Samuel

Question

Determine the equation of line of distance P from the origin

Answer 1

Suppose that the equation of the line is given $Ax+By+C=0$ , and the distance from the origin $O(0;0)$ to the straight line is P

$d=\frac{|Ax_0 +By_0+C|}{\sqrt{A^2+B^2}}, x_0=0, y_0=0, d=P\\$ , then

$$P=\frac{|C|}{\sqrt{A^2+B^2}},\\ C=\pm P \sqrt{A^2+B^2},$$

then the equation of the line at the distance p from the origin is given by the formula

$Ax+By\pm P \sqrt{A^2+B^2}=0$

  where A, B arbitrary and $A^2+B^2 \neq 0 .$