Question

\( \text { True or False? } \) Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The lines represented by \(a x+b y=c_{1}\) and \(b x-a y=c_{2}\) are perpendicular. Assume \(a \neq 0\) and \(b \neq 0\).

Short answer

False. The lines represented by \(ax + by = c1\) and \(bx - ay = c2\) are not perpendicular because the product of their slopes is 1, not -1.

Step-by-step solution

Determine the slopes of the lines

The slope of a line given in the form \(ax + by = c\) is \(-a/b\). Therefore, the slope of the line \(ax + by = c1\) is \(-a/b\), and the slope of the line \(bx - ay = c2\) is \(b/(-a)\).

Check for perpendicularity

The lines are perpendicular if and only if the product of their slopes is -1. Multiplying the two slopes, we get the following: \((-a/b) * (b/(-a))\). This simplifies to \(1\), not \(-1\), so the lines are not perpendicular.