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Find the coordinates of points on the li...

Find the coordinates of points on the line joining the point that is twice as far from as from

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Let `P(x, 3x+3)` be a point on `3x-y+3=0` (as this satisfies the given equation) which is twice as far from `A(3, 2)` as from `B(1, 1)`
So, `" "PA=2PB`
`rArr" "sqrt((x-3)^(2)+(3x+1)^(2))=2sqrt((x-1)^(2) +(3x+2)^(2))`
Squaring both sides, we have
` " "x^(2)- 6x+ 9+9x^(2)+6x+1=4(x^(2)-2x+1+9x^(2)+12x+4)`
`rArr" "30x^(2) +40x+10=0" "rArr" "3x^(2)+4x+1 =0`
`rArr" "(3x+1)(x+1)=0`
`therefore" "x=-1 " "or" "x=-(1)/(3)`
When `x=- 1 rArr y=3 (-1) +3 =0` `" "|"when " x=-(1)/(3)rArry=3(-(1)/(3))+3=2`
So, required points are `(-1, 0) and (-(1)/(3), 2)`.
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