Question

Prove that the following sets of three lines are concurrent:

3x − 5y − 11 = 0, 5x + 3y − 7 = 0 and x + 2y = 0

Answer 1

Given:
3x − 5y − 11 = 0          ... (1)
5x + 3y − 7 = 0            ... (2)
x + 2y = 0                     ... (3)
Now, consider the following determinant:

\[\begin{vmatrix}3 & - 5 & - 11 \\ 5 & 3 & - 7 \\ 1 & 2 & 0\end{vmatrix} = 3 \times 14 + 5 \times 7 - 11 \times 7 = 0\]

Hence, the given lines are concurrent.