A) \[\left( -\sqrt{2},-\sqrt{3} \right)\]
B) \[\left( 3\sqrt{2},2\sqrt{3} \right)\]
C) \[\left( 2\sqrt{2},3\sqrt{3} \right)\]
D) \[\left( \sqrt{3},\sqrt{2} \right)\]
Correct Answer: C
Solution :
Equation of hyperbola is \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] foci is \[(\pm \,2,0)\]hence \[ae=2,\Rightarrow {{a}^{2}}{{e}^{2}}=4\] \[{{b}^{2}}={{a}^{2}}({{e}^{2}}-1)\] \[\therefore \] \[{{a}^{2}}+{{b}^{2}}=4\] ?. Hyperbola passes through \[\left( \sqrt{2},\sqrt{3} \right)\] \[\therefore \] \[\frac{2}{{{a}^{2}}}-\frac{3}{{{b}^{2}}}=1\] ?. On solving [a] and [b] \[{{a}^{2}}=8\](is rejected) and \[{{a}^{2}}=1\]and \[{{b}^{2}}=3\] \[\therefore \] \[\frac{{{x}^{2}}}{1}-\frac{{{y}^{3}}}{3}=1\] Equation of tangent is \[\frac{\sqrt{2x}}{1}-\frac{\sqrt{3y}}{3}=1\] Hence \[\left( 2\sqrt{2},3\sqrt{3} \right)\]satisfy it.
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