In which ratio is the line segment joining the points (1, −7) and (−19, 23) divided by the line, 2x + 7y − 4 = 0?

Let the line segment joining the points (1, −7) and (−19, 23) is divided by the line, 2x + 7y − 4 = 0, in the ratio k : 1.

$\therefore \frac{AP}{PB}=\frac{k}{1}$

Therefore, co-ordinates of P are $\left(\frac{-19k+1}{k+1},\frac{23k-7}{k+1}\right)$.

It can be observed that P is a point on the line, 2x + 7y − 4 = 0.

$\therefore 2\left(\frac{-19k+1}{k+1}\right)+7\left(\frac{23k-7}{k+1}\right)-4=0$

$\Rightarrow -38k+2+161k-49-4k-4=0$

$\Rightarrow 119k=51$

$\Rightarrow k=\frac{3}{7}$

Thus, the line segment joining the points (1, −7) and (−19, 23) is divided by the line, 2x + 7y − 4 = 0, in the ratio 3:7.