Question

If p is the length of perpendicular from the point 1,1) to the line ax-by+c=o be unity the show that 1/c+1/a-1/b= c/2ab

Answer 1

p is the perpendicular distance from the point (1,1) to the line ax - by + c = 0 .

we know, the distance from the point (r, s) to the line lx + my + n = 0 is
$L=\frac{|lr+ms+n|}{\sqrt{l^2+m^2}}$
so, P = |a.1 - b.1 + c |/√(a² + b²)
1 = |a - b + c |/√(a² + b²)

take square root both sides,
1² = (a - b + c )²/(a² + b²)
a² + b² = a² + b² + c² -2ab - 2bc + 2ca
c² + 2ca = 2ab + 2bc

divide both sides with abc

c²/abc + 2ca/abc = 2ab/abc + 2bc/abc
=> c/ab + 2/b = 2/c + 2/a
=> c/ab = 2/c + 2/a - 2/b
dividing by 2 both sides,
c/2ab = 1/c + 1/a - 1/b

$\text{hence,}\:\boxed{\boxed{ \frac{c}{2ab} = \frac{1}{c} + \frac{1}{a} - \frac{1}{b} }}$