Let R be a rectangle given by the lines x = 0, x = 2 , y = 0 and y...

Given,

$R$ be a rectangle given by the lines $x = 0, x = 2$, $y = 0 and y = 5$

And point A \left(\alpha, 0\right)$and B \left(0, \beta\right)$, \alpha \in \left[0, 2\right]$and \beta \in \left[0, 5\right]$,

Now plotting the diagram we get,

Now given that line segment $AB$ divides the ratio of area in $4: 1$, we get

\dfrac{10 - \dfrac{1}{2}\alpha\beta}{\dfrac{1}{2}\alpha\beta} = \dfrac{4}{1}</p><p>⇒ 20 - \alpha\beta = 4\alpha\beta$</p><p>$⇒ \alpha\beta = 4 . . . . . . \left(1\right)$</p><p>Now using midpoint formula we get,</p><p>$h = \dfrac{\alpha}{2}\text{$and$} \beta =\dfrac{k}{2}$</p><p>Now using equation \left(1\right)$ we get,

$⇒ 4hk = 4$

$⇒ xy = 1$ which is a equation of rectangular hyperbola.