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Question

In each of the following find the value of k, for which the points are collinear.
(i) (7,2),(5,1),(3,k)
(ii) (8,1),(k,4),(2,5)

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Solution

(i) let the given points are A(7,2),B(5,1) and C(3,k)
These point are collinear if area (ABC)=0
x1(y2y3)+x2(y3y1)+x3(y1y2)=0
Here x1,y1)=(7.2)
(x2,y2)=(5,1)
(x3,y3)=(3,k)
7(1k)+5(k+2)+3(21)=0
77k+5k+109=0
82k=0
2k=8
k=4
Hence the given points are collinear for k=4

(i) let the given points are A(8,1),B(k,4) and C(2,5)
These point are collinear if area (ABC)=0
x1(y2y3)+x2(y3y1)+x3(y1y2)=0
Here x1,y1)=(8,1)
(x2,y2)=(k,4)
(x3,y3)=(2,5)
8(4+5)+k(51)+2(1+4)=0
86k+10=0
6k=18
k=3
k=4
Hence the given points are collinear for k=3

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