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Standard Equation of Hyperbola

A triangle ha...

Question

A triangle has its vertices on a rectangular hyperbola, then the orthocentre of the triangle lies on the :

A

parabola

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B

same hyperbola

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C

another hyperbola

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D

director circle

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Solution

The correct option is C same hyperbola
let the vertices of the triangle be $(c / t_{1}, c t_{1})$ , $(c / t_{2}, c t_{2})$ , $(c / t_{3}, c t_{3})$
Then its orthocenter is $(\frac{- c}{t_{1} t_{2} t_{3}}, - c t_{1} t_{2} t_{3})$
Therefore, it lies on the same curve $x y = c^{2}$

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