1.

If x^(2) + y^(2) - 4x - 2y + 5 = 0 and x^(2 + y^(2) - 6x - 4y = 0 are membes of a coaxal system of circles then centre of a point circle in the systems is

Answer»

(-5, -6)
(5, 6)
(3,5)
(-8,-13)

SOLUTION :Given CIRCLES are `S -= x^(2) + y^(2) - 4x - 2y + 5 = 0`
and `S^(1) -= x^(2) + y^(2) - 6x - 4y - 3 = 0`
Equation of coaxial system of circles is `S + lambda L - 0`
`L = S - S^(1) = 0`
`implies 2X + 2y + 8 = 0`
`implies x + y + 4 = 0`
`implies x^(2) + y^(2) - 4x - 2y + 5 + lambda x + lambda y +4 lambda = 0`
`implies x^(2) + y^(2) + x (lambda - 4) + y (lambda - 2) + (5 + 4 lambda) = 0`
centre, `C = [(4 - lambda)/(2), (2 - lambda)/(2)]` and radius, r = 0
`implies (16 - 8lambda + lambda^(2) + 4 - 4 lambda + lambda^(2) - 20 - 16 lambda)/(4) = 0`
`implies 2 lambda^(2) - 28 lambda = 0`
`implies lambda (lambda - 14) = 0`
`implies lambda 0, 14`
`implies` centre `C = ((4)/(2), (2)/(2))` (or) `((-10)/(2), (-12)/(2))`
`:. C = (2,1)` (or) (-5, -6)


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