CircleHardBloom L3
Question
A circle passing through origin $O$ cuts two straight lines $x - y = 0$ and $x + y = 0$ in points $A$ and $B$ respectively. If abscissae of $A$ and $B$ are roots of the equation $x^{2} + ax + b = 0$, then the equation of the given circle is:
Options
A.$x^{2} + y^{2} + ax - by = 0$
B.$x^{2} + y^{2} - x\sqrt{4b - a^{2}} + yb = 0$
C.$x^{2} + y^{2} + ax \pm y\sqrt{a^{2} - 4b} = 0$
D.$x^{2} + y^{2} - ax \pm y\sqrt{a^{2} - 4b} = 0$
Solution
Sol. [Key Concept: A circle through $O$ meeting two lines: the abscissae are roots of the given quadratic, so use sum/product to fix the coefficients.]
$${\left( x - x_{1} \right)\left( x - x_{2} \right) + \left( y - x_{1} \right)\left( y + x_{2} \right) = 0 }{x^{2} + y^{2} - \left( x_{1} + x_{2} \right)x + \left( x_{2} - x_{1} \right)y = 0}$$
$$x^{2} + y^{2} + ax \pm \left( \sqrt{a^{2} - 4b} \right)y = 0$$
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