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Q)

Find the equation of the standard rectangular hyperbola whose centre is $(\large-\frac{1}{2} , -\frac{1}{2})$ and which passes through the point $(1 , \large\frac{1}{4})$ Comment
A)
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• 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:
The centre of the rectangular hyperbola is at $(-\large\frac{1}{2}.-\large\frac{1}{2})$ with asymptotes parallel to the coordinate axes.
Therefore its equation is of the form $(x+\large\frac{1}{2})($$y+\large\frac{1}{2})=$$c^2$
Step 2:
The point $(1,\large\frac{1}{4})$ lies on the rectangular hyperbola.
Therefore $(1+\large\frac{1}{2})(\large\frac{1}{4}+\frac{1}{2})=$$c^2 \Rightarrow c^2=\large\frac{9}{8} The equation of the rectangular hyperbola is (x+\large\frac{1}{2})($$y+\large\frac{1}{2})=$$\large\frac{9}{8}$