Which of the following are correct for properties of rectangular hyperbola
$x y = c^{2}$

Area of triangle formed by tangent to

x y = c^{2}

and its intercept is constant and equal to

c^{2} .

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The slope of the tangent for

x y = c^{2}

is always negative.

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If the normal to the rectangular hyperbola

x y = c^{2}

at the point

(t_{1})

meets the curve again at the point

(t_{2})

, then

t_{1}^{3} t_{2} = - 1.

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If a triangle has its vertices on a rectangular hyperbola, then its orthocenter is

H (\frac{- c}{2 t_{1} t_{2} t_{3}}, - c t_{1} t_{2} t_{3}) .

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Solution

The correct option is C If the normal to the rectangular hyperbola $x y = c^{2}$ at the point $(t_{1})$ meets the curve again at the point $(t_{2})$ , then $t_{1}^{3} t_{2} = - 1.$
Option A) The slope of the tangent and slope of the normal for a rectangular hyperbola $x y = c^{2}$ are always negative and positive respectively.
$\to$ Correct

Option B) Area of triangle formed by tangent to hyperbola $x y = c^{2}$ and its intercept is constant and equal to $2 c^{2} .$
$\to$ Incorrect

Option C) If the normal to the rectangular hyperbola $x y = c^{2}$ at the point $(t_{1})$ meets the curve again at the point $(t_{2})$ , then $t_{1}^{3} t_{2} = - 1.$
$\to$ Correct

Option D) If a triangle has its vertices on a rectangular hyperbola, then its orthocenter is $H (\frac{- c}{t_{1} t_{2} t_{3}}, - c t_{1} t_{2} t_{3}) .$
$\to$ Incorrect

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