How to "move" a circle with $r=1$ and defined by $ax^2 + by^2 = q^2 +cx$ over the x-axis

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Is it possible to have an equation in the form of $$ax^2 + by^2 = q^2 +cx$$ or $$f(x)=\sqrt{\frac{q^2+cx-ax^2}{b}}$$

such that

$$f(k)=0; f(m)=0$$

where the distance between $(m,0)$ & $(k,0)$ is 1 and where $m,k>1$ and the area under the curve above the x-axis is $\frac{\pi}{2}$?

In other words, I want to know if it's possible to move a function of a circle with radius $r=1$ over the x-axis