JEE Advanced Model Paper 11 with solutions for online practice

Consider the hyperbola H : x2−y2 = 0 and a circle S with centre C (α, 0).Suppose H and S touch each other at P (θ , Φ) with ϕ > 1 and Φ > 0 . The commontangent to H and S at P intersects the x-axis at point Q. If (β , r) is centroid ofΔPQC then the correct expression (s) among the following is ( are) _________