The circle drawn with variable chord $x + a y - 5 = 0$ (a being a parameter) of the parabola $y ^ { 2 }= 20 { x }$ as diameter will always touch the line

The endpoints of the variable chord of the parabola y^2 = 20x are given by the standard parametric form as (at1^2, 2at1) and (at2^2, 2at2), where 4a = 20 so a = 5. The equation of the chord x + ay - 5 = 0 can be rewritten using the standard chord equation ty = x + at^2. The endpoints of the diameter of the circle are the ends of this focal chord (since it passes through the focus by satisfying the focal chord condition), so the circle passes through the focus (5, 0). The circle drawn on a focal chord as diameter always touches the directrix of the parabola. For the parabola y^2 = 20x, the directrix is the vertical line x + 5 = 0, so the circle touches x + 5 = 0.

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