JEE Main 2023 — Hyperbola Question with Solution

Given,

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

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

$\frac{10-\frac{1}{2}\alpha \beta }{\frac{1}{2}\alpha \beta }=\frac{4}{1}$

$\Rightarrow 20-\alpha \beta =4\alpha \beta$

$\Rightarrow \alpha \beta =4 ......\left(1\right)$

Now using midpoint formula we get,

$h=\frac{\alpha }{2} \& \beta =\frac{k}{2}$

Now using equation $\left(1\right)$ we get,

$\Rightarrow 4hk=4$

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