Question

The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus.

Answer 1

Let P, Q, and R be the points at which the tangents are drawn and let their coordinates be

The tangents at Q and R intersect in the point

Similarly, the other pairs of tangents meet at the points

Let the equation to the circle be

x² + y² + 2gx + 2fy + c = 0 .....(1)

Since it passes through the above three points, we have

Subtracting (3) from (2) and dividing by a(t₂ - t₁), we have

Similarly, from (3) and (4), we have

From these two equations we have

Substituting these values in (2), we obtain

The equation to the circle is therefore

which clearly goes through the focus (a, 0).