21. Determine the ratio in which the line 3x + y - 9 = 0 divides the segment joining the points
(1, 3) and (2, 7).
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Answer:
Using the section formula, if a point (x,y) divides the line joining the points (x
1
,y
1
) and (x
2
,y
2
) in the ratio m:n, then (x,y)=(
m+n
mx
2
+nx
1
,
m+n
my
2
+ny
1
)
Let the ratio be k:1
Substituting (x
1
,y
1
)=(1,3) and (x
2
,y
2
)=(2,7) in thesection formula, we get the point which divides as (
k+1
k(2)+1(1)
,
k+1
k(7)+1(3)
)=(
k+1
2k+1
,
k+1
7k+3
)
Since this point lies on the line 3x+y−9=0, we have
3(
k+1
2k+1
)+
k+1
7k+3
−9=0
=>6k+3+7k+3−9k−9=0
4k−3=0
k=
4
3
Hence, the ratio is a:b=3:4 internally.
So a+b=7
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