How good are you in Ellipse

Geometry Level 5

From a point on the axis of x common tangent are drawn to the parabola \[ y^{2} = 4x \] and the ellipse \[ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\] \((a>b>0) \) . If tangent form an equilateral triangle with their chord of contact with respect to parabola, then set of exhaustive values of a is: \[( \frac{p}{q}, r)\] Find \[( p + q + r) \]

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