Answer:
[tex](x+1)^2+(y+2)^2=25[/tex]
Center = (-1, -2)
Radius = 5
Step-by-step explanation:
Given equation:
[tex]x^2+2x+y^2+4y=20[/tex]
Add the square of half the coefficient of the terms in x and y to both sides of the equation:
[tex]\implies x^2+2x+\left(\dfrac{2}{2}\right)^2+y^2+4y+\left(\dfrac{4}{2}\right)^2=20+\left(\dfrac{2}{2}\right)^2+\left(\dfrac{4}{2}\right)^2[/tex]
Simplify:
[tex]\implies x^2+2x+1+y^2+4y+4=20+1+4[/tex]
[tex]\implies x^2+2x+1+y^2+4y+4=25[/tex]
We have now created two perfect square trinomials for each variable on the left side of the equation:
[tex]\implies (x^2+2x+1)+(y^2+4y+4)=25[/tex]
Factor the perfect square trinomials:
[tex]\implies (x+1)^2+(y+2)^2=25[/tex]
[tex]\boxed{\begin{minipage}{4 cm}\underline{Equation of a circle}\\\\$(x-h)^2+(y-k)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(h, k)$ is the center. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]
Compare the factored equation with the formula for the equation of a circle to find the center and the radius.
Center
[tex]-h=1 \implies h=-1[/tex]
[tex]-k=2 \implies k=-2[/tex]
Therefore, the center is (-1, -2).
Radius
[tex]r^2=25 \implies r=\sqrt{25}=5[/tex]
Therefore, the radius is 5.
