Question

Prove that two different circle cannot intersect each other more than two points

Answer 1

let x^2 + y^2 = a^2
and (x -1)^2 + y^2 = b^2
equation of two different circles .

x^2 + y^2 = a^2 -------(1)

x^2 -2x + 1 + y^2 = b^2 -----------(2)

solve both equation ,

(x^2 + y^2 ) -2x +1 = b^2

a^2 -2x + 1 = b^2

(a^2 -b^2 + 1 )/2 = x

now put x = (a^2 -b^2 + 1)/2 in equation (1)

y^2 = a^2 - {a^2 -b^2 + 1 }^2/4

you see y gain two values let h and k

hence , intersecting points of these two circles are { (a^2- b^2 + 1)/2 , h} and { (a^2-b^2+1)/2 , k}

hence, two circles intersect maximum two points