Circle Tangent to a Parabola?

Geometry Level 5

The above picture shows the graphs of the parabola \(y = x^{2}\) and the line \(y = -\frac{1}{2}x\) in Cartesian coordinates. A circle with center \((p, 0)\), where \(p > 0\), is tangent to both the parabola and the line. What is the value of \(p\) rounded to three decimal places?


Graph generated in Desmos.

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