**Q** Find the equation of the hyperbola with centre as the origin, transverse axis along x - axis, ecentricity √5 and the sum of whose semi - axes is 9.

and the transverse axis is along x-axis, i.e. y =0

therefore let the equation of the hyperbola be $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1..............\left(1\right)$

given : $e=\sqrt{5}anda+b=9$

${b}^{2}={a}^{2}({e}^{2}-1)\phantom{\rule{0ex}{0ex}}\frac{{b}^{2}}{{a}^{2}}=5-1=4\phantom{\rule{0ex}{0ex}}\frac{b}{a}=2\phantom{\rule{0ex}{0ex}}b=2a\phantom{\rule{0ex}{0ex}}a+2a=9\phantom{\rule{0ex}{0ex}}3a=9\Rightarrow a=9/3=3$

b= 2*3=6

thus the required equation of the hyperbola is

$\frac{{x}^{2}}{{3}^{2}}-\frac{{y}^{2}}{{6}^{2}}=1\phantom{\rule{0ex}{0ex}}\frac{{x}^{2}}{9}-\frac{{y}^{2}}{36}=1$

which is the required equation of the hyperbola.

hope this helps you.

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