The equation of the circle passing through (1,2) and the point of intersection of the circles
` x^(2) + y^(2) - 8x - 6y + 21 = 0 " and " x^(2) + y^(2) - 2x - 15 = 0 ` is

Options

A. `x^(2) + y^(2) - 6x - 2y + 9 = 0`
B. `x^(2) + y^(2) + 6x - 2y + 9 = 0`
C. `x^(2) + y^(2) - 6x -4y + 9 = 0`
D. `x^(2) + y^(2) - 6x + 4y + 9 = 0`

Answer

C

Explanation

Given circle are, `S = x^(2) + y^(2) - 8x - 6y + 21 = 0` and `S. -= x^(2) + y^(2) - 2x - 15 = 0` point p (1,2)
`implies` Equaton of radical axis is `S. - S^(1) = 0`
`implies -6x - 6y + 36 = 0`
`implies L -= x + y - 6 = 0`
`implies` Equation of circle passing through point of intersection of S and `S^(1)` is,
`implies S + lambda L = 0`
`implies (x^(2) + y^(2) - 8x - 6y + 21) + lambda (x + y - 6) = 0`
Equation (1) passes through p (1,2)
`implies 1 + 4 - 8 - 12 + 21 + lambda (1 + 2 - 6) = 0`
`implies 3 lambda = 6`
`implies lambda = 2`
Equation (1) `implies x^(2) + y^(2) - 8x - 6y + 21 + 2x + 2y - 12 = 0`
`:. x^(2) + y^(2) - 6x - 4y + 9 = 0`, is required circle