Question

Point A 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 on the line 3x+k(y+1) = 0, find the value of k.

Answer 1

Correct Answer - 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) **** on the line 3x +k(y+1) = 0, we have
$(3 xx 2) +k(-4 +1) = 0 rArr 3k = 6 rArr k = 2.$