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Question
Prove that the following sets of three lines are concurrent:
\[\frac{x}{a} + \frac{y}{b} = 1, \frac{x}{b} + \frac{y}{a} = 1\text { and } y = x .\]
Solution
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.
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