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Question 11
Find the ratio in which the point
P(34,512)
divides the line segment joining the points A(12,32) and B(2, -5).

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Solution

Let P(34,512) divide AB internally in the ratio m:n.
Using the section formula, we get
(34,512)=(2mn2m+n,5m+32nm+n)
[ By internal section formula, the coordinates of point P dividing the line segment joining the point (x1,y1) and (x2,y2) in the ratio m1:m2 internally is
(m2x1+m1x2m1+m+2,m2y1+m1y2m1+m+2)]On equating, we get
34=2mn2m+n and 512=5m+32nm+n34=4mn2(m+n) and 512=10m+3n2(m+n)32=4mnm+n and 56=10m+3nm+n3m+3n=8m2n and 5m+5n=60m+18n5n5m=0 and 65m13n=0n=m and 13(5mn)=0m =n does not satisfy.Since, m =n does not satisfy.5mn=05m=nmn=15Hence, the required ratio is 1:5.

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