x^2 +4x +9y^2 -36y +4=0
This is the equation of an ellipse.
I assume you want it in standard form
complete the square
x^2 + 4x + 4 + 9(y^2 - 4y + 4) = -4 + 4 + 36
(x+2)^2 + 9(y-2)^2 = 36
divide both sides by 36
(x+2)^2 /36 + (y-2)^2 /4 = 1
The given equation is:
x^2 + 4x + 9y^2 - 36y + 4 = 0
To understand the type of equation we have here, we can look at the form of the equation. It appears to have quadratic terms (x^2 and y^2) and linear terms (4x and -36y), as well as a constant term (4). This suggests that it might be a conic section equation.
To further analyze the equation, we can rewrite it by grouping the x terms and the y terms:
(x^2 + 4x) + (9y^2 - 36y) + 4 = 0
Now let's factor the quadratic terms:
x(x + 4) + 9y(y - 4) + 4 = 0
At this point, we can see that we have a parabolic term (x^2) and an elliptical term (9y^2), since the coefficients are positive and different. The constant term (4) suggests there might be a circle involved.
To determine the specific conic section, we usually compare the coefficients of x^2 and y^2. By comparing 1 (coefficient of x^2) with 9 (coefficient of y^2), we can see that they are different. Thus, this equation represents an ellipse.
So, the equation x^2 + 4x + 9y^2 - 36y + 4 = 0 represents an ellipse.
However, if you have specific requirements or constraints for the equation, please provide more details so that a more accurate analysis can be performed.