Area of triangle formed by tangent to the hyperbola $x y = 16$ at $(16, 1)$ and coordinate axes

8

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16

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32

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64

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Solution

The correct option is C $32$
Differenciating $x y = 16, \frac{x d y}{x} + y = 0$
$\Rightarrow \frac{d y}{d x} = \frac{- y}{x} = \frac{- 1}{16} =$ Slope of tangent
$\Rightarrow$ Its (tangent's) equation : $y = - 1 = - \frac{1}{16} (x - 16)$
$\Rightarrow 16 y - 16 = - x + 16 \Rightarrow 16 y = - x - 32$
$\Rightarrow 16 y + x + 32 = 0$
It will cut $x -$ axis at $A (- 32, 0)$
& $y -$ axis at $B (0, - 2)$
$\Rightarrow$ Area of $△ O A B = \frac{1}{2} \times 2 \times 32 \Rightarrow (C)$

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