Find the foot of the perpendicular through the point A ( − 3 ; 2 ) to line 2 x − y + 4 = 0

Question

Find the foot of the perpendicular through the point   A  (  −  3  ;  2  ) to line   2  x  −  y  +  4  =  0

Athena Blanchard · Accepted Answer

Step 1Any point on the line   2  x  −  y  +  4  =  0 is of the form   (  t  ∈      R    )Then slope of the line   A  N  :      m    1    =            (      2      t      +      4      )      −      2              t      −      (      −      3      )        =            2      t      +      2              t      +      3      Now slope of the line   2  x  −  y  +  4  =  0  :      m    2    =  2Two lines are perpendicular, hence       m    1    ⋅      m    2    =  −  1            2      t      +      2              t      +      3        ⋅  2  =  −  1    ⟹    t  =            −      7        5  Hence,   N  =  (            −      7        5    ,      6    5    )

Thaddeus Sanders · Accepted Answer

Step 1We have   y  =  2  x  +  4 . Consider the line throudh A that is perpendicular to this line.This line hence must have slope   −      1    2   .Thus its represented by   y  =  −      1    2    x  +  b for some b.But since it passesthrough A,   2  =      3    2    +  b    ⟹    b  =      1    2   .Thus the line is represented by   y  =  −      1    2    x  +      1    2   . The foot of the perpendicular is simply the intersection of the two lines. Setting the y-coordinates, equal:  2  x  +  4  =      1    2    −      1    2    x    ⟹    4  x  +  8  =  1  −  x    ⟹    x  =  −      7    5  And then   y  =  −      14    5    +  4  =      6    5   . So the answer is       (    −          7      5        ,          6      5        )   .

Find the foot of the perpendicular through the point

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