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Math Help - Intersection of Lines problem

  1. #1
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    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

    thx in advance for answering
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  2. #2
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    Quote Originally Posted by helpmeinmath View Post
    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}}.
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