The lengthk of the equation chord of the two circles `(X - a)^(2) + y^(2) = a^(2)` and `x^(2) + (y - b)^(2) = b^(2)` is

Options

A. `(ab)/(sqrt(a^(2) + b^(2)))`
B. `(2 ab)/(sqrt(a^(2) + b^(2)))`
C. `(a + b)/(sqrt(a^(2) + b^(2)))`
D. `sqrt(a^(2) + b^(2))`

Answer

B

Explanation

Given circle are, `S -= x^(2) + y^(2) - 2ax = -` and `S^(1) = x^(2) + y^(2) - 2by = 0`
`implies C_(1) (a, 0)` and `r_(1 = sqrt(a^(2) + 0 - 0) = a`
`implies ` Radical axis is, `S - S^(1) = 0`
`implie - 2ax + 2by = 0`
`implies ax - by = 0`
d - Perpendicular distance from `C_(1)` to ax - by = 0
`implies d = |(a^(2) - 0 + 0)/(sqrt(a^(2) + b^(2)))| = (a^(2))/(sqrt(a^(2) + b^(2)))`
`:.` Length of the chord `= sqrt(r_(1)^(2) - d^(2))`
`= 2 sqrt(a^(2) - (a^(4))/(a^(2) + b^(2)))`
`= 2 sqrt((a^(4) + a^(2) b^(2) - a^(4))/(a^(2) + b^(2)))`
`= (2ab)/(sqrt(a^(2) + b^(2)))`