Question

If the hyperbolas, $x^2+3xy+2y^2+2x+3y+2=0$ and $x^2+3xy+2y^2+2x+3y+c=0$ are conjugate of each other, the value of $c$ is equal to

• A. $-2$
• B. $4$
• C. $0$
• D. $1$

Answer 1

Answer: (C)

$0$

The given hyperbola is $x^2+3xy+2y^2+2x+3y+2=0$ ...(1)

We already know that the equation of the asymptote of a hyperbola differs from the hyperbola by a constant.

$\therefore$ Let $x^2+3xy+2y^2+2x+3y+k=0$ ...(2)

be the equation of the asymptotes of the given hyperbola.

Hence, equation (2) must represent a pair of straight lines, the condition for which is

$abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

$\Rightarrow \left(1\right)\left(2\right)\left(k\right)+2\left(\frac{3}{2}\right)\left(1\right)\left(\frac{3}{2}\right)-1(\frac{3}{2}{)}^2-2(1{)}^2-k(\frac{3}{2}{)}^2=0$

$\Rightarrow 2k+\frac{9}{2}-\frac{9}{4}-2-\frac{9}{4}k=0$

$\Rightarrow k=1$

Therefore, the asymptotes are given by $x^2+3xy+2y^2+2x+3y+1=0$ .

The equation of conjugate hyperbola is $2A-H=0$

where, $A$ is the equation of asymptotes

$H$ is the equation of given hyperbola

$2(x^2+3xy+2y^2+2x+3y+1)-$$(x^2+3xy+2y^2+2x+3y+2)=0$

$\therefore x^2+3xy+2y^2+2x+3y=0$

Hence, $c=0$.