# Concurrency of Three Lines

## Concurrency of Three Lines

Three linesare said to be concurrent if they pass through a common point, i.e. they meetat a point.

Thus, if three lines are concurrent the point of intersection of the two lines on the third line. Let the three concurrent lines be

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

a₃x + b₃y + c₃ = 0

Then thepoint of intersection of (i) and (ii) must lie on the third

The coordinate of the point of intersection of (i) and (ii) are

$$\left( \frac{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}},\frac{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}} \right)$$.

This point lies on line (iii), therefore we get

$${{a}_{3}}\left( \frac{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}} \right)+{{b}_{3}}\left( \frac{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}} \right)+{{c}_{3}}=0$$.

(or)

$$\left|\begin{matrix} {{a}_{1}} &{{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} &{{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} &{{c}_{3}} \\\end{matrix} \right|=0$$.

Example: Find the value k, if the lines 3x – 4y – 13 = 0, 8x – 11y – 33 and 2x – 3y + k = 0 are concurrent.

Solution: Given that

3x- 4y – 13 = 0,

8x -11y-33 and

2x – 3y + k = 0

The given lines are concurrent if

$$\left|\begin{matrix} {{a}_{1}} &{{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} &{{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} &{{c}_{3}} \\\end{matrix} \right|=0$$.

$$\left|\begin{matrix} 3 & -4 & 13 \\ 8& -11 & -33 \\ 2 & -3 & k \\\end{matrix} \right|=0$$.

3 (- 11k – 99) – 4 (8k + 66) + 13 (- 24 + 22) = 0

– k – 7 = 0

K = – 7