Question

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 that point A also lies on the line 3x + k( y + 1 ) = 0, find the value of k.

Answer 1

Let the coordinates of A be`(x,y) Here  (PA)/(PQ) = 2/5 . so ,`

PA + AQ= PQ

`⇒PA +AQ =(5PA)/2               [∵ PA = 2/5 PQ]`

` ⇒AQ = (5PA)/2 - PA`

`⇒ (AQ)/(PA) = 3/2 `

`⇒ (PA)/(AQ) = 2/3 `

Let (x, y) be the coordinates of A, which dives PQ in the ratio 2 : 3 internally Then using section formula, we get

` X = (2 xx (-4) +3 xx (6))/(2+3) = (-8+18)/5= 10/5 = 2`

`y = (2 xx (-1) + 3 xx(-6))/(2+3) = (-2-18)/5 = (-20)/5 = -4`

Now, the point ( 2, -4 )  lies on the line 3x +k(y+1) = 0 ,therefore

3 × 2 +k(-4+1)=0

⇒ 3k = 6

`⇒ k =6/3 =2`

Hence, k=2.