Equation $x^{2} + 2 a x - b^{2} = 0$ has real roots $α, β$ and equation $x^{2} + 2 p x - q^{2} = 0$ has real roots $γ, δ$ If circle C is drawn with the points $(α, γ), (β, δ)$ as extremities of a diameter, then the equation of C is

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$x^{2} + y^{2} + 2 a x + 2 p y - b^{2} - q^{2} = 0$

$x^{2} + y^{2} + 2 a x + 2 n v + b^{2} + a^{2} = 0$

$x^{2} + y^{2} - 2 a x - 2 p y + b^{2} + q^{2} = 0$

$x^{2} + y^{2} + 2 a x - 2 p y + b^{2} - q^{2} = 0$

answer is A.

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Detailed Solution

$α, β$ are the roots of $x^{2} + 2 a x - b^{2} = 0 \Rightarrow (x - α) (x - β) = x^{2} + 2 a x - b^{2}$

$γ, δ$ are the roots of $x^{2} + 2 p x - q^{2} = 0 \Rightarrow (x - γ) (x - δ) = x^{2} + 2 p x - q^{2}$

$\Rightarrow (y - γ) (y - δ) = y^{2} + 2 p y - q^{2}$

The equation of the circle is $(x - α) (x - β) + (y - γ) (y - δ) = 0$

$\Rightarrow x^{2} + 2 a x - b^{2} + y^{2} + 2 p y - q^{2} = 0 \Rightarrow x^{2} + y^{2} + 2 a x + 2 p y - b^{2} - q^{2} = 0$

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