Question

In what ratio, the line joining (-1, 1) and (5, 7) is divided by the line x + y = 4?

Answer 1

The equation of the line joining the points (-1, 1) and (5, 7) is given by

\(y-1=\frac{7-1}{5+1}(x+1)\)

\(y-1=\frac{6}{6}(x+1)\)

\(x-y+2=0 .....(1)\)

The equation of the given line is 
\(x + y - 4 = 0 … (2) \)
The 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 ratio \(1:k.\)
Accordingly, by section formula,

\((1,3)=\left(\frac{k(-1)+1(5)}{1+k},\frac{k(1)+1(7)}{1+k}\right)\)

\(⇒ (1,3)=\left(\frac{-k+5}{1+k},\frac{k+7}{1+k}\right)\)

\(⇒\frac{ -k+5}{1+k}=1,\frac{k+7}{1+k}=3\)

\(\frac{-k+5}{1+k}=1\)

\(⇒ -k+5=1+k\)
\(⇒ 2k=4\)
\(⇒ k=2\)
Thus, the line joining the points (-1, 1) and (5, 7) is divided by line \( x + y = 4\) in the ratio \(1:2\).