Answer:
Step-by-step explanation:
In order to do this, you will need to complete the square on both the x terms and the y terms. Begin by grouping the x's and the y's together (and they already are). Next, just for organizational purposes, put the x's together into a set of ( ), and do the same with the y terms:
[tex](x^2+2x)+(y^2+4y)=20[/tex]
The process here is to take half the linear terms, square them, and then add them to both sides. This is very simple, and the process is the same every time,whether you are using this to do what we're doing with a circle, whether we are factoring a quadratic, or whether we are using it to write a parabola in vertex form.
Our linear x term is 2 (from 2x; just the 2, not the x). Half of 2 is 1, and 1 squared is 1, so we will add a 1 into the ( ) and onto the other side (to keep the equation in balance and all of that); the linear term on the y term is 4. Half of 4 is 2, and 2 squared is 4, so we will add a 4 into the ( ) and onto the other side:
[tex](x^2+2x+1)+(y^2+4x+4)=20+1+4[/tex]
Clean up the right side to get 25. That's easy.
The reason we complete the square is because, during this process, we have ensured that we have a perfect square binomial. The ( ) with the x's has a perfect square binomial of
[tex](x+1)^2[/tex] (which gives us the h of the center of the circle), and the ( ) with the y's has a perfect square binomial of
[tex](y+2)^2[/tex] (which gives us the k of the center of the circle!). The radius is the square root of the constant on the right, which is 5. All in all, here's our circle in standard form:
[tex](x+1)^2+(y+2)^2=25[/tex]
That means that the center is (-1, -2) {yes the signs are the opposite due to the standard form of the equation}, and the radius is 5. That is choice 3 down for you.
The center is located at (−1, −2), and the radius is 5.
The circle equation is given as:
x^2 + 2x + y^2 + 4y = 20
Complete the square on x and y.
So, we have:
x^2 + 2x + (2/2)^2+ y^2 + 4y + (4/2)^2 = 20 + (2/2)^2 + (4/2)^2
Evaluate the exponents
x^2 + 2x + 1+ y^2 + 4y + 4 = 20 +1 + 4
Expand
x^2 + x + x + 1+ y^2 + 2y + 2y + 4 = 25
Factorize
x(x + 1) + 1(x + 1)+ y(y + 2) + 2(y + 2) = 25
Factor out x + 1 and y + 2
(x + 1)^2 + (y + 2)^2 = 25
Express 25 as 5^2
(x + 1)^2 + (y + 2)^2 = 5^2
The equation of a circle is represented as:
(x - a)^2 + (y - b)^2 = r^2
Where:
Center = (a,b)
Radius = r
So, we have:
Center = (-1,-2)
Radius = 5
Hence, the center is located at (−1, −2), and the radius is 5.
Read more about circle equations at:
https://brainly.com/question/1506955
