Two circles each of radius 5 units touch each other at the point ( 1 , 2 ) . If the equation of their common tangent is 4 x + 3 y = 10 , and C 1 ( α , β ) and C 2 ( γ , δ ) , C 1 ≠ C 2 are their centres, then | ( α + β ) ( γ + δ ) | is equal to

Step-by-step solution

Let a point circle: ( x - 1 ) 2 + ( y - 2 ) 2 = 0 Equation of circle passing through the intersection of the line and the circle: C 1 + λ L 1 = 0 ( x - 1 ) 2 + ( y - 2 ) 2 + λ ( 4 x + 3 y - 10 ) = 0 x 2 + y 2 + ( 2 λ - 1 ) 2 x + 3 2 λ - 2 2 y + 5 - 10 λ = 0 r = g 2 + f 2 - c r = ( 2 λ - 1 ) 2 + 3 2 λ - 2 2 - ( 5 - 10 λ ) = 5 4 λ 2 + 1 - 4 λ + 9 4 λ 2 + 4 - 6 λ - 5 + 10 λ = 25 25 4 λ 2 - 25 = 0 λ = ± 2 for λ = 2 x 2 + y 2 + 6 x + 2