A line perpendicular to the line segment joining the points $(1, 0)$ and $(2, 3)$ divides it in the ratio $1 : n$. Find the equation of the line.

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Accepted Answer

> [!TLDR]
> The required line is perpendicular to the segment joining $(1,0)$ and $(2,3)$, so its slope is $-\frac{1}{3}$. It passes through the point that divides the segment in the ratio $1:n$, which is $\left(\frac{n+2}{n+1}, \frac{3}{n+1}ight)$. The equation is $x + 3y = \frac{n+2}{n+1} + \frac{9}{n+1} = \frac{n+11}{n+1}$.

The key idea here is that a line perpendicular to a given segment has a slope that is the negative reciprocal of the segment's slope. Once we know the slope, we only need one point on the line to write its equation — and that point is given by the division ratio condition.

Let’s work through it step by step.

1. **Find the slope of the given segment.**  
   The segment joins $A(1,0)$ and $B(2,3)$.  
   Slope $m_{AB} = \frac{3 - 0}{2 - 1} = 3$.

2. **Find the slope of the perpendicular line.**  
   If two lines are perpendicular, the product of their slopes is $-1$.  
   So if the required line has slope $m$, then $m 	imes 3 = -1$, giving $m = -\frac{1}{3}$.

> [!FORMULA]
   > Perpendicular slopes condition: $m_1 \cdot m_2 = -1$.

3. **Find the point where the line cuts the segment.**  
   The line divides $AB$ in the ratio $1 : n$. This means it passes through a point $P$ on $AB$ such that $AP : PB = 1 : n$.

Using the section formula: if a point divides the join of $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m : n$ (measured from the first point), the coordinates are  
   $$ \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} ight). $$

Here $m = 1$, $n = n$ (careful: the ratio is $1 : n$, so the first part is 1, the second is $n$).  
   So $P = \left( \frac{1 \cdot 2 + n \cdot 1}{1 + n}, \frac{1 \cdot 3 + n \cdot 0}{1 + n} ight) = \left( \frac{2 + n}{n + 1}, \frac{3}{n + 1} ight)$.

> [!WARNING]
   > A common mistake is to swap the ratio. If the ratio is $1 : n$, then the first coordinate uses $1$ times the second point and $n$ times the first point — not the other way around. Always check which point is fi

Q.A line perpendicular to the line segment joining the points $(1, 0)$ and $(2, 3)$ divides it in the ratio $1 : n$ . Find the equation of the line.

Concept understanding — Perpendicular Slopes Condition

Perpendicular Slopes Condition

The Intuition

The Precise Statement

What About Vertical and Horizontal Lines?

Quick Check

Why It Works (A Short Proof)

Q.A line perpendicular to the line segment joining the points (1,0)(1, 0)(1,0) and (2,3)(2, 3)(2,3) divides it in the ratio 1:n1 : n1:n. Find the equation of the line.

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Q.A line perpendicular to the line segment joining the points $(1, 0)$ and $(2, 3)$ divides it in the ratio $1 : n$ . Find the equation of the line.

Unlock everything free for 14 days