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Question

Find the coordinates of the points which divide the line segment joining A(2,2) and B(2,8) into four equal parts.

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Solution

Let P,Q,R be the points which divide the line into 4 equal parts

Then the ratio of AP and PB=m1,m2=1:3

Here A(x1,y1)=A(2,2),B(x2,y2)=B(2,8)

Co-ordinates of P=(m1x2+m2x1m1+m2,m1y2+m2y1m1+m2)

(1×22×31+3,1×8+3×21+3)

(264,8+64)=(1,72)

The ratio of AQ and QB=m1:m2=2:2

Here A(x1,y1)=A(2,2),B(x2,y2)=B(2,8)

Co-ordinates of Q=(m1x2+m2x1m1+m2,m1y2+m2y1m1+m2)

(2×2+2×21+3,2×8+2×21+3)

(4+44,16+44)=(0,5)

Then the ratio of AR and RB=m1:m2=3:1

Here A(x1,y1)=A(2,2),B(x2,y2)=B(2,8)

Co-ordinates of R=(m1x2+m2x1m1+m2,m1y2+m2y1m1+m2)

(3×2+1×21+3,3×8+1×21+3)

(624,24+24)=(1,132)

Coordinates are P(1,72),(0,5),(1,132)

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