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Question

What are the coordinates of the foot of perpendicular from the point (1,3) to the line 3x-4y-16=0?


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Solution

Step 1. Obtain the equations required to find coordinates of point.

Let us consider the coordinates of the foot of the perpendicular from (1,3) to the line 3x-4y-16=0 be (a,b).

So, the slope of the line join (-1,3) and (a,b) is m=b-3a+1

And the slope of the line 3x-4y-16=0 or y=34x-4 is m'=34.

Since these two lines are perpendicular,

Thus m×m'=-1

b-3a+1×34=-13b-94a+4=-13b-9=-4a-44a+3b=5...1

The point a,b lies in the line 3x-4y=16

So, 3a-4b=16...2

Step 2. Find the coordinates of the foot of perpendicular.

In order to solve (1) and (2) , multiply the equation (1) with 4 and equation (2) with 3 and add both the equations:

16a+12b+9a-12b=20+4825a=68a=6825

Put a=6825 in equation (2):

3×6825-4b=1620425-4b=16-4b=16-20425-4b=16×25-20425-4b=400-20425-4b=19625b=19625×-4b=-4925

Hence a,b=6825,-4925

Hence, the coordinates of the foot of perpendicular are6825,-4925.


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