Question

prove that the perpendicular drawn from the point (4,1) on the join of (2, - 1) and (6,5 ) divides it in the ratio 5 ratio 8

Answer 1

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Question:

Prove that the perpendicular drawn from the point (4, 1) on the join of (2, – 1) and (6 5) divides it in the ratio 5:8.

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)=6k+1/2 ,5k-1/ k+1.............(i)

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

The slope of AB = 5+1/6-2

The slope of CR = y-1/x-4

And, PR is perpendicular to AB

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

(y-1/x-4)×(5+1/6-2)= -1

{6k+2/k+1} -1/{5k-1/k+1}-4 ×6/4 = -1

5k-1-k-1/6k-2-4k-4= -4/6

3(4k – 2) = – 2(2k – 2)

3(4k – 2) = – 2(2k – 2)12k – 6 = – 4k + 4

3(4k – 2) = – 2(2k – 2)12k – 6 = – 4k + 416k = 10

K=5/8

So, The ratio is 5:8

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

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