Conic-Sections Question 590
Question: Equation of the hyperbola whose directirx is $ 2x+y=1 $ , focus (1, 2) and eccentricity $ \sqrt{3} $ is
Options:
A) $ 7x^{2}-2y^{2}+12xy-2x+14y-22=0 $
B) $ 5x^{2}-2y^{2}+10xy+2x+5y-20=0 $
C) $ 4x^{2}+8y^{2}+8xy+2x-2y+10=0 $
D) None of these
Show Answer
Answer:
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 $ $ {{{ \frac{2x+y-1}{\sqrt{4+1}} }}^{2}}(\because e=\sqrt{3}) $
$ \Rightarrow 5(x^{2}+y^{2}-2x-4y+5) $ $ =3(4x^{2}+y^{2}+1+4xy-2y-4x) $
$ \Rightarrow 7x^{2}-2y^{2}+12xy-2x+14y-22=0 $ Which is the required hyperbola.
BETA
