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Q)

Find the equation of the asymptotes of the following rectangular hyperbola $xy-kx-hy=0$

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A)
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  • http://clay6.com/mpaimg/2_toolbox%2021.jpg
  • For the standard rectangular hyperbola,the coordinate axes are taken to the asymptotes and the equation of the hyperbola is $xy=c^2$ .
  • If the centre is at $(h,k)$ with asymptotes parallel to the coordinate axes,the equation is $(x-h)(y-k)=c^2$
Step 1:
$xy-kx-hy=0$
The combined equation of the asymptotes differs only in the constant term.
Now $xy-kx-hy=xy-kx-hy+kh-kh$
$\Rightarrow x(y-k)-h(y-k)-kh$
$\Rightarrow (x-h)(y-k)-kh$
Therefore the equation of the rectangular hyperbola is $(x-h)(y-k)-kh=0$
$(x-h)(y-k)=kh$
Step 2:
This is of the form $(x-h)(y-k)=c^2$ and is an rectangular hyperbola with centre $(h,k)$ and asymptotes $x-h=0,y-k=0$
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