A) \[{{x}^{2}}-16xy-11{{y}^{2}}-12x+6y+21=0\]
B) \[3{{x}^{2}}+16xy+15{{y}^{2}}-4x-14y+1=0\]
C) \[{{x}^{2}}+16xy+11{{y}^{2}}-12x+6y+21=0\]
D) none of the above
Correct Answer: A
Solution :
Key Idea: If P be any point on the hyperbola, S be a focus and M be a point on the directrix, then \[PS=ePM.\] Since, \[S=(2,1),e=2\] Equation of directrix is \[x+2y=1,\] \[\because \] \[PS=ePM\] \[\therefore \] \[{{(x-2)}^{2}}+{{(y-1)}^{2}}=4{{\left( \frac{x+2y-1}{\sqrt{{{1}^{2}}+{{2}^{2}}}} \right)}^{2}}\] \[\Rightarrow \] \[5({{x}^{2}}+{{y}^{2}}-4x-2y+5)\] \[\Rightarrow \] \[=4({{x}^{2}}+4{{y}^{2}}+1+4xy-4y-2x)\] \[\Rightarrow \] \[{{x}^{2}}-11{{y}^{2}}-16xy-12x+6y+21=0\]
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