Question

If the length of the perpendicular from the point (1, 1) to the line ax − by + c = 0 be unity, show that \[\frac{1}{c} + \frac{1}{a} - \frac{1}{b} = \frac{c}{2ab}\] .

 

Answer 1

The distance of the point (1, 1) from the straight line ax − by + c = 0 is 1 

\[\therefore 1 = \left| \frac{a - b + c}{\sqrt{a^2 + b^2}} \right|\]
\[ \Rightarrow a^2 + b^2 + c^2 - 2ab + 2ac - 2bc = a^2 + b^2 \]
\[ \Rightarrow ab + bc - ac = \frac{c^2}{2}\]

Dividing both the sides by abc, we get:

\[\frac{1}{c} + \frac{1}{a} - \frac{1}{b} = \frac{c}{2ab}\]