If two chords of the circle $x^{2} + y^{2} - ax - by = 0$, drawn from the point $(a,b)$ is divided b

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If two chords of the circle $x^{2} + y^{2} - ax - by = 0$, drawn from the point $(a,b)$ is divided by the $x$-axis in the ratio $2:1$ then:

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[Key Concept: Two chords from $(a,b)$ on the circle through the origin: combine the section ratio with power of a point.]Put $\left( h, - \frac{b}{2} \right)$ to equation of circle$$h^{2} + \frac{b^{2}}{4} - ah + \frac{b^{2}}{2} = 0 $$$h^{2} - ah + \frac{3b^{2}}{4} = 0$ must get two distinct real roots$$\Rightarrow D > 0 \Rightarrow a^{2} - 3b^{2} > 0$$

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