x²+4y²+2x-8y+1=0
What about it,
looks like you have an ellipse.
I will assume you want to find the properties after completing the square
x^2 + 2x + ... + 4y^2 - 8y + .... = -1 + .....
x^2 + 2x + 1 + 4(y^2 - 2y + 1) = -1 + 1 + 4
(x + 1)^2 + 4(y - 1)^2 = 4
divide each term by 4
(x+1)^2 / 4 + (y - 1)^2 = 1
draw any conclusions from this
The given equation is a quadratic equation in terms of x and y. It can be rewritten in the standard form of a conic section, which is given by:
Ax² + By² + Cx + Dy + E = 0
Comparing this with the given equation, we have:
A = 1
B = 4
C = 2
D = -8
E = 1
To find out what type of conic section the equation represents, we can calculate the discriminant:
Discriminant (D) = B² - 4AC
Plugging in the values, we get:
D = 4² - 4(1)(4) = 16 - 16 = 0
Since the discriminant is zero, the equation represents a parabola.
To determine the specific equation of the parabola, we need to complete the square for both the x and y terms.
Starting with the x-terms:
x² + 2x
To complete the square, add and subtract the square of half the coefficient of x:
(x² + 2x + 1) - 1
Now let's do the same for the y-terms:
4y² - 8y
(x² + 2x + 1) - 1 + 4(y² - 2y + 1) - 1 = 0
Combining like terms, we get:
(x² + 2x + 1) - 1 + 4(y² - 2y + 1) - 1 = 0
(x + 1)² - 1 + 4(y - 1)² - 1 = 0
(x + 1)² + 4(y - 1)² - 2 = 0
Therefore, the simplified equation of the parabola is:
(x + 1)² + 4(y - 1)² = 2