If P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3) and | vedclass

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(C) For a rectangular hyperbola $xy = c^2$,any point on it can be represented as $(ct_i, c/t_i)$.Let the vertices of $\Delta PQR$ be $P(ct_1, c/t_1)$,

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$(-x_4, -y_4)$

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(C) For a rectangular hyperbola $xy = c^2$,any point on it can be represented as $(ct_i, c/t_i)$.Let the vertices of $\Delta PQR$ be $P(ct_1, c/t_1)$,$Q(ct_2, c/t_2)$,and $R(ct_3, c/t_3)$.The orthocentre $H$ of $\Delta PQR$ is given by the coordinates $\left( \frac{-c}{t_1t_2t_3}, -c(t_1t_2t_3) \right)$.Since the four points $P, Q, R, S$ are concyclic on the hyperbola $xy = c^2$,the product of their parameters satisfies $t_1t_2t_3t_4 = 1$,which implies $t_1t_2t_3 = 1/t_4$.Substituting this into the orthocentre coordinates,we get $H = \left( \frac{-c}{1/t_4}, -c(1/t_4) \right) = (-ct_4, -c/t_4)$.Since $S(x_4, y_4) = (ct_4, c/t_4)$,the orthocentre is $(-x_4, -y_4)$.

If $P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3)$ and $S(x_4, y_4)$ are $4$ concyclic points on the rectangular hyperbola $xy = c^2$,the coordinates of the orthocentre of the triangle $PQR$ are:

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