The combined equation of the asymptotes of the hyperbola `2x^2+5x y+2y^2+4x+5y=0` is a. `2x^2+5x y+2y^2+4x+5y+2=0` b.`2x^2+5x y+2y^2+4x+5y-2=0` c.`2x^2+5x y+2y^2=0` d.none of these

Options

A. `2x^(2)+5xy+2y^(2)+4x+5y+2=0`
B. `2x^(2)+5xy+2y^(2)+4x+5y-2=0`
C. `2x^(2)+5xy+2y^(2)=0`
D. none of these

Answer

A

Explanation

Let the equation of the asymptotes be
`2x^(2)+5xy+2y^(2)+4x+5y+lambda=0" (1)"`
This equation represents a pair of straight lines. Therefore,
`abc+2fgh-af^(2)-bg^(2)-ch^(2)=0`
We have
`4lambda+25-(25)/(2)-8-lambda(25)/(4)=0`
`"or "=(9lambda)/(4)+(9)/(2)=0`
`"or "lambda=2`
Putting the value of `lambda` in (1), we get
`2x^(2)+5xy+2y^(2)+4x+5y+2=0`
This is the equation of the asymptotes.