A) \[7{{x}^{2}}-2{{y}^{2}}+12xy-2x+14y-22=0\]
B) \[5{{x}^{2}}-2{{y}^{2}}+10xy+2x+5y-20=0\]
C) \[4{{x}^{2}}+8{{y}^{2}}+8xy+2x-2y+10=0\]
D) None of these
Correct Answer: A
Solution :
[a] Let \[P(x,y)\] be any point on the hyperbola and PM is perpendicular form P on the directrix, Then by definition, \[SP=ePM\] \[\Rightarrow {{(SP)}^{2}}={{e}^{2}}{{(PM)}^{2}}\] \[\Rightarrow {{(x-1)}^{2}}+{{(y-2)}^{2}}=3\] \[{{\left\{ \frac{2x+y-1}{\sqrt{4+1}} \right\}}^{2}}(\because e=\sqrt{3})\] \[\Rightarrow 5({{x}^{2}}+{{y}^{2}}-2x-4y+5)\] \[=3(4{{x}^{2}}+{{y}^{2}}+1+4xy-2y-4x)\] \[\Rightarrow 7{{x}^{2}}-2{{y}^{2}}+12xy-2x+14y-22=0\] Which is the required hyperbola.
You need to login to perform this action.
You will be redirected in
3 sec
