Question

Does `4x^2+4xy-y^2-6x-3y-4=0` represents two straight lines?

Answer 1

Please see below.

Explanation:

If `4x^2+4xy-y^2-6x-3y-4=0` represents two straight lines,

it should be possible to factorize it as `(ax+by+c)(lx+my+n)=0`

We can write `4x^2+4xy-y^2-6x-3y-4=0` as

`4x^2+2x(2y-3)-(y^2+3y+4)=0`

If the equation represents a pair of lines, it should be possible to write `x` as one or other of two linear expressions in `x` and `y` and so the square root of discriminant must be rational.

Now discriminant is `4(2y-3)^2+16(y^2+3y+4)`

= `4(4y^2-12y+9)+16(y^2+3y+4)`

= `32y^2+100`

But this is not so. Hence it does not represent a pair of strainght lines. The equation actually reppresents a hyperbola.

graph{4x^2+4xy-y^2-6x-3y-4=0 [-20, 20, -10, 10]}

Observe that for `4x^2+4xy+y^2-6x-3y-4=0`, we get discriminant as `100` and then it represents a pair of lines.

In fact `4x^2+4xy+y^2-6x-3y-4=0` can be written as

`(2x+y+1)(2x+y-4)=0` or `2x+y+1=0` and `2x+y-4=0`, a pair of parallel lines as shown below.

graph{4x^2+4xy+y^2-6x-3y-4=0 [-20, 20, -10, 10]}