Tangents AB and AC are drawn to the circle `x^2 +y^2-2x + 4y +1=0` from `A(0.1)` then equation of circle passing through `A,B and C` is

Options

A. `x^(2) + y^(2) + x+ y - 2 = 0`
B. `x^(2) + y^(2) - x+ y - 2 = 0`
C. `x^(2) + y^(2) + x - y - 2 = 0`
D. None of these

Answer

B

Explanation

Equation of chord of contact (BC) T = 0
`x(0) + y(1) -2 ((x+0)/(2)) + 4 ((y+1)/(2)) + 1 = 0 `
`implies -x + 3y + 3 = 0 `
Let equation of circle passes through A , B , C is `S + lambda L = 0 `
`implies (x^(2) + y^(2) - 2x + 4y + 1) + lambda (-x + 3y + 3) = 0 `
It passes through A (0,1)
`therefore ( 0 + 1 - 0 + 4 + 1) + lambda (0 + 3 + 3)= 0`
`lambda = -1 `
so equation of the required circle is
`x^(2) + y^(2) -x + y - 2 = 0`