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In the text book that I have, a question asked:- $$x^{2}+y^2+6x-2+\lambda(x^2+y^2-9x+2y+1)=0$$ For which value(s) of $\lambda$ does the above equation represents a point circle?

What was to be done was simple, arrange the equation in the standard form of a circle and then equate radius to zero.

BUT, one thing that I couldn't understand in this is, that the above equation represents the family of circles passing through the points of intersection of the individual circles. How can a point circle pass through two points(to look at the equation from a different perspective)?

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  • $\begingroup$ It's possible only if the 2 circles touch each other, which is not the case. $\endgroup$ – CY Aries May 14 '17 at 11:45
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Its always good to have intuition about the given situation. You correctly figured out that no such $\lambda$ can exist, which satisfies the given condition (how can a point pass through two points when the two points are distinct).

If we blindly go about solving for radius, we form a quadratic in $\lambda$, and find that the discriminant is negative, suggesting no such $\lambda$ exists.

$$r^2=\left(3-9\frac{\lambda}{2}\right)^2 + \lambda^2-\lambda+2=0$$

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  • $\begingroup$ So, the answer would be complex numbers... done, thanks... $\endgroup$ – Gaurav Agarwal May 14 '17 at 11:49

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