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Point A lies on the line segment PQ join...

Point A lies on the line segment PQ joining P(6, -6) and Q(-4, -1) in such a way that `(PA)/(PQ) = (2)/(5)`. If the point A also lies on the line 3x+k(y+1) = 0, find the value of k.

Text Solution

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The correct Answer is:
k = 2

`(PA)/(PQ) = (2)/(5) rArr (PQ)/(PA) = (5)/(2) rArr (PA +AQ)/(PA) = (5)/(2)`
`rArr 1 + (AQ)/(PA) = (5)/(2) rArr (AQ)/(PA) = ((5)/(2)-1) = (3)/(2)`
`rArr (PA)/(AQ) = (2)/(3) rArr PA: AQ = 2:3`
`therefore "coordinates of A are"((2 xx (-4) + 3 xx 6)/(2+3), (2 xx (-1) + 3 xx (-6))/(2+3)) = A(2, -4).`
Since the point A(2, -4) lies on the line 3x +k(y+1) = 0, we have
`(3 xx 2) +k(-4 +1) = 0 rArr 3k = 6 rArr k = 2.`
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