Find the foot of the perpendicular through the point

Alissa Hutchinson 2022-05-07 Answered
Find the foot of the perpendicular through the point A ( 3 ; 2 ) to line 2 x y + 4 = 0
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Answers (2)

Athena Blanchard
Answered 2022-05-08 Author has 8 answers
Step 1
Any 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 = 2
Two 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 )
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Thaddeus Sanders
Answered 2022-05-09 Author has 1 answers
Step 1
We 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 ) .
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