Question

2073 Set D Q.NO. 13 If the length of the perpendicular from the
point (1.1) to the line ax + by - C = 0 is 1, show that
1/a +1/b –1/c = c/2ab​

Answer 1

Step-by-step explanation:

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}}L=

l

2

+m

2

∣lr+ms+n∣

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} }}hence,

2ab

c

=

c

1

+

a

1

−

b

1