Q.A line is such that its segment between the lines $5x - y + 4 = 0$ and $3x + 4y - 4 = 0$ is bisected at the point $(1, 5)$. Obtain its equation.

Section Formula (Vector Form)

Given two points, where is the point that divides the segment joining them in a chosen ratio? The section formula answers this with position vectors, generalising the midpoint to any ratio.

Setup

Let $P$ and $Q$ have position vectors $\vec{a}$ and $\vec{b}$ (measured from the origin $O$). We want the position vector $\vec{r}$ of the point $R$ that divides $PQ$ in the ratio $m:n$, i.e. $PR:RQ=m:n$.

Internal division

When $R$ lies between $P$ and $Q$:

Notice the cross-pairing: the far endpoint $Q$ (position $\vec{b}$) is weighted by $m$, and the near endpoint $P$ (position $\vec{a}$) by $n$. The result is a weighted average of the endpoints, so $R$ sits closer to whichever endpoint carries the larger opposite weight.

Midpoint as a special case

Put $m=n$ (ratio $1:1$):

the familiar midpoint formula. So the section formula is just a generalised midpoint.

External division

When $R$ lies on the line $PQ$ but outside the segment (say beyond $Q$), the denominator changes sign:

Why it matters …