In what ratio does the line \(x+y=4\) divide the segment joining the points \((-1,\,1)\) and \((5,\,7)\)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 1 : 2
Let the line cut the segment in the ratio \(k:1\). The section formula gives the cutting point in terms of \(k\), and that point must satisfy the equation of the line.
Point dividing the segment from \((-1,\,1)\) to \((5,\,7)\) in the ratio \(k:1\): \(\displaystyle \left(\frac{5k-1}{k+1},\,\frac{7k+1}{k+1}\right)\)
It lies on \(x+y=4\): \(\displaystyle \frac{5k-1}{k+1}+\frac{7k+1}{k+1}=4\)
\(\displaystyle \frac{12k}{k+1}=4\), so \(12k=4k+4\)
\(8k=4\), so \(\displaystyle k=\frac{1}{2}\)
Ratio \(\displaystyle =\frac{1}{2}:1=1:2\)
Check: the cutting point is \(\displaystyle \left(\frac{5-2}{3},\,\frac{7+2}{3}\right)=(1,\,3)\), and \(1+3=4\)
