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 $1:2$.