Question

Prove that the following sets of three lines are concurrent:

\[\frac{x}{a} + \frac{y}{b} = 1, \frac{x}{b} + \frac{y}{a} = 1\text {  and } y = x .\]

Answer 1

Given: 

\[bx + ay - ab = 0\]           ... (1)

\[ax + by - ab = 0\]           ... (2)

x − y = 0                   ... (3)

Now, consider the following determinant:

\[\begin{vmatrix}b & a & - ab \\ a & b & - ab \\ 1 & - 1 & 0\end{vmatrix} = - b \times ab - a \times ab - ab \times \left( - a - b \right) = 0\]

Hence, the given lines are concurrent.