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 point P also lies on the line segment 3x + k(y+1) = 0 then find the value of k
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PA/PQ=2/5
or, PA/(PA+AQ)=2/5
or, 5PA=2PA+2AQ
or, 5PA-2PA=2AQ
or,3PA=2AQ
or, PA/AQ=2/3
or, m:n=2:3
If the coordinates of A is (x,y) the,
x=(mx₂+nx₁)/(m+n) and y=(my₂+ny₁)/(m+n)
Here, (x₁,y₁)=(6,-6), (x₂,y₂)=(-4,-1)
∴, x={2×(-4)+3×6)}/(2+3)
=(-8+18)/5
=10/5
=2
y={2×(-1)+3×(-6)}/(2+3)
=(-2-18)/5
=-20/5
=-4
Now, p(2,-4) lies on 3x+k(y+1)=0 then
3×2+k(-4+1)=0
or, 6+k(-3)=0
or, -3k=-6
or, k=2 Ans.
or, PA/(PA+AQ)=2/5
or, 5PA=2PA+2AQ
or, 5PA-2PA=2AQ
or,3PA=2AQ
or, PA/AQ=2/3
or, m:n=2:3
If the coordinates of A is (x,y) the,
x=(mx₂+nx₁)/(m+n) and y=(my₂+ny₁)/(m+n)
Here, (x₁,y₁)=(6,-6), (x₂,y₂)=(-4,-1)
∴, x={2×(-4)+3×6)}/(2+3)
=(-8+18)/5
=10/5
=2
y={2×(-1)+3×(-6)}/(2+3)
=(-2-18)/5
=-20/5
=-4
Now, p(2,-4) lies on 3x+k(y+1)=0 then
3×2+k(-4+1)=0
or, 6+k(-3)=0
or, -3k=-6
or, k=2 Ans.
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