Question

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 rectangle hyperbola $xy=c^2$, then coordinates of the orthocentre of the triangle $\Delta PQR$ is :

• A. $(x_4,-y_4)$
• B. $(x_4,y_4)$
• C. $(-x_4,-y_4)$
• D. $(-x_4,y_4)$

Answer 1

The correct option is C $(-x_4,-y_4)$

A rectangular hyperbola circumscribing a triangle also passes through its orthocentre.
If $(c{t}_i,\frac{c}{t_i})$, where $i=1,2,3,$ are the vertices of the triangle.
Then the orthocenter is $(\frac{-c}{t_1{t}_2{t}_3},-c{t}_1{t}_2{t}_2)$, where $t_1{t}_2{t}_3{t}_4=1$.
Hence, orthocentre is $(-c{t}_4,\frac{-c}{t_4})=(-x_4,-y_4)$.