Find the condition, if the two lines ax + by = c and a′x + b′

1.	Find the condition, if the two lines ax + by = c and a′x + b′y = c′ are perpendicular.
Answer» The given lines are ax + by = c ⇒ y = \(-\frac{ax}{b}+\frac{c}{b},m_1=-\frac{a}{b}\) and a'x +b'y = c' ⇒ y = \(\frac{a'}{b'}x+\frac{c'}{b'},m_2=-\frac{a'}{b'}\) Since the line are perpendicular. ∴ m₁m₂ = – 1 ⇒ (−\(\frac{a}{b}\) ) (−\(\frac{a'}{b'}\)) = − 1 ⇒ \(\frac{aa'}{bb'}=-1\) ⇒ aa′ + bb′ = 0, which is the required condition.

Find the condition, if the two lines ax + by = c and a′x + b′y = c′ are perpendicular.

Answer»

The given lines are

ax + by = c ⇒ y = \(-\frac{ax}{b}+\frac{c}{b},m_1=-\frac{a}{b}\)

and a'x +b'y = c' ⇒ y = \(\frac{a'}{b'}x+\frac{c'}{b'},m_2=-\frac{a'}{b'}\)

Since the line are perpendicular.

∴ m₁m₂ = – 1

⇒ (−\(\frac{a}{b}\) ) (−\(\frac{a'}{b'}\)) = − 1

⇒ \(\frac{aa'}{bb'}=-1\)

⇒ aa′ + bb′ = 0, which is the required condition.