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Distance between Parallel Lines

Let the equat...

Question

Let the equation of tangent at the point $(1, 2)$ to the rectangular hyperbola $x y = 2$ be $a x + y = k$ .Find $k + a$ ?

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Solution

$x y = 2 \Rightarrow y = \frac{2}{x}$
$\Rightarrow \frac{d y}{d x} = - \frac{2}{x^{2}}$
$∴$ slope of tangent at $(1, 2)$ is, $m = {(\frac{d y}{d x})}_{(1, 2)} = - \frac{2}{x^{2}} = - 2$
Hence equation of the tangent is, $(y - 2) = - 2 (x - 1)$
$\Rightarrow 2 x + y = 4$
Thus the value of $k + a$ is $6$

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