Through a point $P(-2,0)$ tangents $PQ$ and $PR$ are drawn to the parabola $y^2=8x.$ Two circles each passing through the focus of the parabola and one touching at Q and the other at R are drawn. Which of the following point(s) w.r.t. the $\Delta PQR$ lie(s) on the common chord of the two circles?

Geometry Problem: Parabola and Circles

Given:

• Point P(-2, 0).
• Tangents PQ and PR drawn to parabola y² = 8x.
• Two circles drawn, each passing through the focus of the parabola S(2, 0): one touches at Q, the other at R.

Key geometric fact:

The common chord of the two circles passing through Q, S and R, S is the radical axis. This radical axis also contains points related to ΔPQR.

Conclusion:

The points P, Q, R, and the parabola's focus S all lie on the common chord (radical axis) of the two circles. Therefore, with respect to the triangle ΔPQR, the point P and the focus S lie on the common chord of the two circles.