Intersection of Lines problem

Quote:

Originally Posted by helpmeinmath

Part A
Find the point G at which the normal through the origin intersects the line Ax + By + C = 0 in the x y plane

Part B
Find the magnitude of the position vector OG of point G

The normal to the line Ax + By + C = 0 is in the direction of the vector (A,B). So G will be a point of the form (x,y) = (rA,rB), that lies on the line. Therefore $rA^2+rB^2+C=0$ , and thus $r = \frac{-C}{A^2+B^2},$ $G = \Bigl(\frac{-AC}{A^2+B^2}, \frac{-BC}{A^2+B^2}\Bigr).$

It's then easy to check that $|OG| = \frac{|C|}{\sqrt{A^2+B^2}}.$