# Locus if three lines are concurrent

If the three distinct lines

\begin{align} x + 2 a y + a & = 0 \\ x + 3 b y + b & = 0 \\ x + 4 a y + a & = 0 \end{align}

are concurrent, then the point (a, b) lies on a :-

(1) circle (2) straight line (3) parabola (4) hyperbola

I try to solve two equation of line and put the values of x ,y in third equation . but it is getting very long , is there any other method .

I think there would be some method to solve it with matrix

• By first & third we have $ay=0$ – lab bhattacharjee Mar 18 '17 at 12:51
• @labbhattacharjee that's a nice one – hey Mar 18 '17 at 12:55

For the system of equations to have solution, the matrix of coefficients must have vanishing determinant: $$\begin{vmatrix} 1 & 2a & a \\ 1 & 3b & b \\ 1 & 4a & a \\ \end{vmatrix}=0.$$