Question

1/x - 1/y varies 1/x-y ,prove that X2 + Y2 varies xy​

Answer 1

Given that,

$\longrightarrow\left(\dfrac{1}{x}-\dfrac{1}{y}\right)\propto\dfrac{1}{x-y}$

Let,

$\longrightarrow\dfrac{1}{x}-\dfrac{1}{y}=\dfrac{k}{x-y}$

where k is a non - zero independent constant.

Then,

$\longrightarrow\dfrac{1}{x}-\dfrac{1}{y}=\dfrac{k}{x-y}$

Making LHS into a single fraction,

$\longrightarrow\dfrac{y-x}{xy}=\dfrac{k}{x-y}$

Multiplying by -1,

$\longrightarrow\dfrac{x-y}{xy}=\dfrac{-k}{x-y}$

By cross multiplication,

$\longrightarrow(x-y)^2=-kxy$

Expanding LHS,

$\longrightarrow x^2+y^2-2xy=-kxy$

Adding $2xy$ to each side,

$\longrightarrow x^2+y^2=(2-k)xy$

Let $2-k$ be a non - zero (independent) constant such that $k\in\mathbb{R}-\{0,\ 2\}.$

Then we get,

$\longrightarrow\underline{\underline{x^2+y^2\propto xy}}$

Hence Proved!

Answer 2

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refer attachment

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