Line formed by joining (-1, 1) and (5.7) is divided by a line x + y 4 in the
ratio of
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In what ratio, the line joining (−1,1) and (5,7) is divided by the line x+y=4 ?
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Suppose the line x+y=4 divides the join of A(−1,1) and B(5,7) in the ratio λ:1.
The coordinates of the point of division are (
λ+1
5λ+1
,
λ+1
7λ+1
). It lies on x+y=4.
Therefore,
λ+1
5λ−1
+
λ+1
7λ+1
=4
5λ−1+7λ+1=4(λ+1)
12λ=4λ+4
8λ=4
λ=
2
1
Hence, the required ratio is 1:2.
Step-by-step explanation:
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Answer:
The equation of the line joining the points (−1,1) and (5,7) is given by,
y−1=
5+1
7−1
(x+1)
y−1=
6
6
(x+1)
x−y+2=0......(1)
and the equation of the given line is
x+y−4=0.....(2)
Thus point of intersection of lines (1) and (2) is given by,
x=1 and y=3
Let point (1,3) divide the line segment joining (−1,1) and (5,7) in the ration 1:k.
Accordingly, by section formula,
(1,3)={
1+k
k(−1)+1(5)
,
1+k
k(1)+1(7)
}
⇒(1,3)={
1+k
−k+5
,
1+k
k+7
}
⇒
1+k
−k+5
=1,
1+k
k+7
=3
⇒
1+k
−k+5
=1
⇒−k+5=1+k
⇒2k=4⇒k=2
Thus, the line joining the points (−1,1) and (5,7) is divided by line x+y=4 in the ratio 1:2.
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