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