Question

For what values of a and b the intercepts cut off on the coordinate axes by the line ax + by + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2x − 3y + 6 = 0 on the axes. 

Answer 1

We have 2x − 3y + 6 = 0

\[\Rightarrow \frac{2}{- 6}x - \frac{3}{- 6}y = \frac{- 6}{- 6}\]

\[ \Rightarrow \frac{x}{- 3} + \frac{y}{2} = 1\]

The x and y intercepts of the above line are −3 and 2 respectively.
Now, ax + by + 8 = 0

\[\Rightarrow \frac{a}{- 8}x + \frac{b}{- 8}y = \frac{- 8}{- 8}\]

\[ \Rightarrow \frac{x}{\frac{- 8}{a}} + \frac{y}{\frac{- 8}{b}} = 1\]

The x and y intercepts of the above line are \[\frac{- 8}{a} \text { and  } \frac{- 8}{b}\] respectively.
According to the question, 

\[\frac{- 8}{a} = - \left( - 3 \right) \text { and } \frac{- 8}{b} = - \left( 2 \right)\]

\[ \Rightarrow a = - \frac{8}{3}\text { and }b = 4\]