Show that the equation `2x^(2) + xy - y^(2) + x + 4y - 3 = 0` represents a pair of lines.

Explanation

Given, `2x^(2)+xy-y^(2)+x+4y-3=0`
On comparing it with
`ax^(2)+2hxy+by^(2)+2gx+2fy+c=0`
We get `a=2,h=(1)/(2),b=-1,g=(1)/(2),f=2,c=-0`
Consider D `=|{:(a,h,g),(h,b,f),(g,f,c):}|`
`=|{:(2,(1)/(2),(1)/(2)),((1)/(2),-1,2),((1)/(2),2,-3):}|`
`=2(3-4)-(1)/(2)(-(3)/(2)-1)+(1)/(2)(1+(1)/(2))`
`=-2+(5)/(4)+(3)/(4)=0`
Since D=0, the given equation repesents a pair of straight lines.