The point (3,-4) lies on both the circles ` x^(2) + y^(2) - 2x + 8y + 13 = 0 ` and ` x^(2) + y^(2) - 4x+ 6y + 11 = 0 ` . Then the angle between the circles is :

Options

A. `60^(@)`
B. `tan^(-1) ((1)/(2))`
C. `tan^(-1) ((3)/(5))`
D. `135^(@)`

Answer

D

Explanation

Given circles are, `S -= x^(2) + y^(2) - 2x + 8y + 13 = 0`
and `S^(1) -= x^(2) + y^(2) - 4x + 6y + 11 = 0` and P (3, - 4)
`implies C_(1) (-1,4)` and `C_(2) = (2, - 3)`
`r_(1) = sqrt(1 + 16 - 13) = 2` and `r_(2) = sqrt(4 + 9 - 11) = sqrt(2)`
Radical axis, `S - S^(1) = 0`
`implies 2x + 2y + 2 = 0`
`implies x + y + 1 = 0`
`implies a^(1) = |(1 - 4 + 1)/(sqrt(1 + 1))| = (2)/(sqrt(2)) = sqrt(2)`
`implies cos theta = (d^(2) - r_(1)^(2) - r_(2)^(2))/(2r_(1)r_(2)) = (2 - 4 - 2)/(2 xx 2 sqrt(2)) = (-4)/(4sqrt(2)) = (1)/(sqrt(2)) = cos 135^(@)`
`:. theta = 135^(@)`