We know that the distance between the centre of the circle and any point on the circumference of the circle is the radius.

We know that the circle $\(C\)$ has centre at the point $\((5,-3)\)$ and cuts the $\(x\)$-axis at the points $\((4,0)\)$ & $\((6,0)\)$, so the radius of the circle $\(C\)$ is the distance between $\((5,-3)\)$ and $\((4,0)\)$ or the distance between $\((5,-3)\)$ and $\((6,0)\)$.

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Now, using the distance formula, we get the radius of the circle C as $\(\sqrt{(5-4)^2+(-3-0)^2}\)$ $\(=\sqrt{1+9}=\sqrt{10}=\mathrm{r}\)$, say


(Distance between points $\(\left(x_1, y_1\right) \&\left(x_2, y_2\right)\)$ is $\(\left.\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}\right)\)$

So, the circumference of the circle C is $\(2 \pi \mathrm{r}=2 \times \pi \times \sqrt{10}=2 \sqrt{10} \pi\)$ (= column A quantity) which is clearly less than $\(10 \pi\)$ ( $\(=\)$ column B quantity).

Hence the answer is (B).