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Equation of Family of Circles Passing through Points of Intersection of Circle and a Line

Find the equa...

Question

Find the equation of the circle passing through the points
$(1, 2) (3, - 4)$ and $(5, - 6)$

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Solution

equation of circle passing through $(1, 2) (3, 4) (5, - 6)$

$x^{2} + y^{2} + 2 h x + 2 k y + c = 0$
$1 + 4 + 2 h + 4 k + c = 0$
$2 h + 4 k + 5 + c = 0 - - - - (1)$
$9 + 16 + 6 h + 8 k + c = 0$
$6 h + 8 k + c + 25 = 0 - - - - (2)$
$25 + 36 + 10 h - 12 k + c = 0$
$10 h - 12 k + c + 61 = 0 - - - - - - (3)$
eq $(1)$ & $(2)$ eq $(1)$ & $(3)$
$(2 h + c + 15 = 0) 4$ $10 h + 12 k + 15 + 3 c = 0 10 h - 12 k + c + 61 = 0_16 h + 4 c + 76 = 0 8 h + 4 c + 60 = 0 -__8 h = 16 h = 2 x^{2} + y^{2} - 4 x + 5 y - 11 = 0$

$- 4 + c + 15 = 0$

$c = - 11$

$2 h + 4 k + 5 + c = 0$

$2 (- 2) + 4 k + 5 - 11 = 0$

$4 k = 10$

$k = 2.5$

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