If the length of the perpendicular from the point (1, 1) to the line ax – by + c = 0 be unity, Show that 1/c + 1/a - 1/b = c/2ab.

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

Given: 

Line ax – by + c = 0 and point (1, 1) 

To prove:

\(\frac{1}{c} + \frac{1}{a} - \frac{1}{b} = \frac{c}{2ab}\) 

Concept Used: 

Distance of a point from a line. 

Explanation: 

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

∴ 1 = \(|\frac{a-b+c}{\sqrt{a^2+b^2}}|\)

⇒ a² + b² + c²– 2ab + 2ac – 2bc = a² + b² 

⇒ ab + bc – ac = \(\frac{c^2}{2}\)

Dividing both the sides by abc, we get:

\(\frac{1}{c} + \frac{1}{a} - \frac{1}{b} = \frac{c}{2ab}\) 

Hence proved.