Question

Find the point of intersection of the following pairs of lines:

bx + ay = ab and ax + by = ab.

Answer 1

The equations of the lines are as follows:
bx + ay = ab   

\[\Rightarrow\] bx + ay − ab = 0         ... (1)
ax + by = ab  

\[\Rightarrow\] ax + by − ab = 0          ... (2)
Solving (1) and (2) using cross-multiplication method:   

\[\frac{x}{- a^2 b + a b^2} = \frac{y}{- a^2 b + a b^2} = \frac{1}{b^2 - a^2}\]

\[ \Rightarrow \frac{x}{ab\left( b - a \right)} = \frac{y}{ab\left( b - a \right)} = \frac{1}{\left( a + b \right)\left( b - a \right)}\]

\[ \Rightarrow x = \frac{ab}{a + b} \text { and } y = \frac{ab}{a + b}\]

Hence, the point of intersection is \[\left( \frac{ab}{a + b}, \frac{ab}{a + b} \right)\].