# Intersection of Lines problem

• Jun 2nd 2008, 01:07 PM
helpmeinmath
Intersection of Lines problem
I am studying for an upcoming exam and i came across this question that many of my classmates didt get so heres the question

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

• Jun 3rd 2008, 12:49 AM
Opalg
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 $\displaystyle rA^2+rB^2+C=0$, and thus $\displaystyle r = \frac{-C}{A^2+B^2},$ $\displaystyle G = \Bigl(\frac{-AC}{A^2+B^2}, \frac{-BC}{A^2+B^2}\Bigr).$

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