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If P(x1,y1),Q(x2,y2),R(x3,y3) and S(x4...

If `P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3) and S(x_4,y_4)` are four concyclic points on the rectangular hyperbola ) and `xy = c^2` , then coordinates of the orthocentre ofthe triangle `PQR` is

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We know that the orthocentre of the triangle formed by points `P(t_(1)), Q(t_(2)) and R(t_(3))` on hyperbola lies on the same hyperbola and has coordinates `((-c)/(t_(1)t_(2)t_(3)),-ct_(1)t_(2)t_(3))`.
Also, if points `P(t_(1)), Q(t_(2)),R(t_(3)) and S(t_(4))` are concyclic then `t_(1)t_(2)t_(3)t_(4)=1.`
Therefore, orhtocentre of triangle PQR is `(-ct_(4),(-c)/(t_(4)))-=(-x_(4),-y_(4)).`
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