If the circle x^2+y^2+2 alpha x+c=0 lies comp | vedclass

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(D) The centre of the circle $C_1: x^2+y^2+2 \alpha x+c=0$ is $O_1(-\alpha, 0)$ and its radius is $r_1 = \sqrt{\alpha^2-c}$.The centre of the circle $

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$\alpha \beta > 0$

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(D) The centre of the circle $C_1: x^2+y^2+2 \alpha x+c=0$ is $O_1(-\alpha, 0)$ and its radius is $r_1 = \sqrt{\alpha^2-c}$.The centre of the circle $C_2: x^2+y^2+2 \beta x+c=0$ is $O_2(-\beta, 0)$ and its radius is $r_2 = \sqrt{\beta^2-c}$.For $C_1$ to lie completely inside $C_2$,the distance between centres $d = |O_1O_2| = |-\alpha - (-\beta)| = |\beta - \alpha|$ must satisfy $d + r_1 Since $r_1 Also,$|\beta - \alpha| Squaring both sides or analyzing the condition for containment,we find that for the circles to be nested with the same constant term $c$,the condition $\alpha \beta > 0$ must hold to ensure the centres are on the same side of the origin and the geometry allows for containment.

If the circle $x^2+y^2+2 \alpha x+c=0$ lies completely inside the circle $x^2+y^2+2 \beta x+c=0$,then which of the following holds?

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