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Equation of Family of Circles Passing through Point of Intersection of Two Circles

Find the equa...

Question

Find the equation of the circle circumscribing the triangle formed by the lines $L_{1} = 0, L_{2} = 0$ and $L_{3} = 0$

$L_{1} + λ L_{2} + μ L_{3} = 0$

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$L_{1} L_{2} + λ L_{2} L_{3} + μ L_{3} L_{1} = 0$

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$L_{1} L_{2} L_{3} = 0$

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$L_{1}^{2} + λ L_{2}^{2} + μ L_{3}^{2} = 0$

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Solution

The correct option is B
$L_{1} L_{2} + λ L_{2} L_{3} + μ L_{3} L_{1} = 0$

Family of circles circumscribing a triangle whose sides are $L_{1} = 0, L_{2} = 0$ and $L_{3} = 0$ is

$L_{1} L_{2} + λ L_{2} L_{3} + μ L_{3} L_{1} = 0$ - - - - - - (1)

It is the definition of family of circle & circumscribing a triangle.

Equation (1) is a equation of circle so, it should follow following conditions

(1) coefficient of $x y = 0$

(2) coefficient of $x^{2} =$ coefficient of $y^{2}$

Option B is correct

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