A) \[~2{{x}^{2}}+\text{ }3xy-2{{y}^{2}}-\text{ }5x\,\,+\,\,5y-8=0\]
B) \[{{x}^{2}}+3xy-2{{y}^{2}}-5x+5y+8=0\]
C) \[2{{x}^{2}}+3xy-2{{y}^{2}}+5x-5y+8=0\]
D) None of these
Correct Answer: A
Solution :
The given equation of hyperbola is \[2{{x}^{2}}+3xy-2{{y}^{2}}-5x+5y+2=0\] ?(i) The equation of hyperbola conjugate to the hyperbola (i) is given by, hyperbola + conjugate hyperbola = 2 (asymptotes) ... (ii) Let the equation of asymptotes of hyperbola (i) is \[2{{x}^{2}}+3xy-2{{y}^{2}}-5x+5y+\lambda =0\] ?(iii) Comparing this equation with general equation of second degree \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0,\]we get \[a=2,h=\frac{3}{2},b=-2,\,g=-\frac{5}{2},f=\frac{5}{2},c=\lambda \] Eq. (iii) will represent a pair of straight lines (asymptotes), if \[abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0\] \[\Rightarrow \]\[(2)(-2)\lambda +2\left( \frac{5}{2} \right)\left( -\frac{5}{2} \right)\left( \frac{3}{2} \right)-2{{\left( \frac{5}{2} \right)}^{2}}\] \[-(-2){{\left( -\frac{5}{2} \right)}^{2}}-\lambda {{\left( \frac{3}{2} \right)}^{2}}=0\] \[\Rightarrow \] \[-4\lambda -\frac{75}{4}-\frac{50}{4}+\frac{50}{4}-\frac{9}{4}\lambda =0\] \[\Rightarrow \] \[\frac{25}{4}\lambda =-\frac{75}{4}\] \[\Rightarrow \] \[\lambda =-3\] \[\therefore \] Equation of asymptotes is \[2{{x}^{2}}+3xy-2{{y}^{2}}-5x+5y-3=0\] Now, putting the equations of hyperbola and asymptotes in Eq. (i), we get the equation of conjugate hyperbola is 2 (asymptotes) - hyperbola = 0 \[\Rightarrow \]\[2(2{{x}^{2}}+3xy-2{{y}^{2}}-5x+5y-3)\] \[-(2{{x}^{2}}+3xy-2{{y}^{2}}-5x+5y+2)=0\] \[\Rightarrow \]\[2{{x}^{2}}+3xy-2{{y}^{2}}-5x+5y-8=0\]
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