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Show that the three points A(2,3,4),\ B(...

Show that the three points `A(2,3,4),\ B(-1,2,-3)a n d\ C(-4,1,-10)` are collinear and find the ratio in which C divides AB.

Text Solution

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Given points are A ( 2,3,4) B (-1,2,-3) and C (-4,1,10)
`AB=sqrt((2+1)^(2)+(3+2)^(2)+(4+3)^(2))`
= `sqrt(9+1+49)=sqrt59`
` BCsqrt((-1+4)^(2)+(2-1)^(2)+ (-3+10)^(2))`
`sqrt(9+1+49)=sqrt59`
`AC = sqrt((2+4)^(2)+(3-1)^(2)+(4+10)^(2))`
`sqrt(36+4+196)`
`sqrt236 = 2sqrt59`
Now, `AB+BC= sqrt59+sqrt59=2sqrt59`
AB + BC =AC
Hence, the point A ,B, and C are collinear
Now, AC: BC=`2sqrt59: sqrt59=2:1`
So, C divide AB in 2:1 externally.
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