Answer :

Given, A perpendicular drawn from the point (4,1) on the join of (2, – 1) and (6,5)

To Prove: The perpendicular divides the line in the ratio 5:8.



Explanation: Let us Assume,The perpendicular drawn from point C(4,1) on a line joining A(2, – 1) and B(6,5) divide in the ratio k:1 at the point R.


Now, The coordinates of R are:


By using Sectional Formula, (x,y) =


R(x,y) = – – – (1)


The slope of the line with two points is, m =


The slope of AB =


The slope of CR =


And, PR is perpendicular to AB


Since, (Slope of CR)×(Slope of AB) = – 1






3(4k – 2) = – 2(2k – 2)


12k – 6 = – 4k + 4


16k = 10


K


So, The ratio is 5:8


Hence, R divides AB in the ratio 5:8.


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