show that the equation x^2+2xy+y^2+x+y-2=0 rebresent two paralil lines and find the equations of thos lines
x^2+2xy+y^2+x+y-2=0
Decomposition:
(x+y-1)(x+y+2)=0
Expresion has zeros when:
x+y-1=0
and
(x+y+2)=0
x+y-1=0
y= -x+1
x+y+2=0
y= -x-2
Equation of sraight line:
y = mx + b
m=slope
b=y-intercept
y= -x+1 Slope -1
y= -x-2 Slope -1
Lines with the same slope are parallel.
To show that the equation represents two parallel lines, we need to rewrite the given equation in the standard form of a second-degree equation (ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0), where the coefficients satisfy certain conditions.
First, let's rearrange the given equation:
x^2 + 2xy + y^2 + x + y - 2 = 0
Now, let's group the x^2, xy, and y^2 terms together, as well as the x and y terms together:
(x^2 + 2xy + y^2) + (x + y) - 2 = 0
(x + y)^2 + (x + y) - 2 = 0
Now, notice that we have a quadratic equation in terms of (x + y). Let's introduce a new variable, u = (x + y), which allows us to rewrite the equation as:
u^2 + u - 2 = 0
This quadratic equation can be factorized as follows:
(u + 2)(u - 1) = 0
So, we have two possible cases:
Case 1: u + 2 = 0
Solving for u, we get:
u = -2
Case 2: u - 1 = 0
Solving for u, we get:
u = 1
Now, substitute back the values of u into the equation x + y = u:
For Case 1: -2
x + y = -2
y = -2 - x
For Case 2: 1
x + y = 1
y = 1 - x
Therefore, the equations of the two parallel lines represented by the given equation are:
y = -2 - x
y = 1 - x