Answer :
Answer:
The center of the circle is at (-2, 3) and has a radius of [tex]\sqrt{13}[/tex].
Step-by-step explanation:
Circles
The standard equation for a circle is
[tex](x-h)^2+(y-k)^2=r^2[/tex],
where (h, k) is the center of the circle and r is its radius.
[tex]\dotfill[/tex]
Completing the Square
When a square is completed, the quadratic can be factored where the factors are the same and can thus be written as a single factored squared.
For example,
[tex]x^2+10x+25=(x+5)(x+5)=(x+5)^2[/tex].
All completed squares should end as [tex](x-s)^2[/tex] for any real number s. This should look familiar to the circle equation.
To complete the square, we must add a new "c" value. To find that "c" value, take the squared value of one-half of b.
Make sure that when you add it into the quadratic it's also added to the other side of the equation.
I.e.
[tex]ax^2+bx+d+\left(\dfrac{b}{2} \right)^2= \left(\dfrac{b}{2} \right)^2[/tex],
where d doesn't make a complete square.
All that there's left is to factor between the x squared term, the bx, and the new c value we added.
[tex]\hrulefill[/tex]
Solving the Problem
We need to complete the square twice, for the x and y terms, for our initial equation to have the standard circle form.
We can start by organizing the terms.
[tex]x^2+4x+y^2-6y-3=0[/tex]
We can even bring the 3 onto the other side to make things easier.
[tex]x^2+4x+y^2-6y=3[/tex]
We use the general formula to find the new c value for the x and y terms:
[tex]c_x=\left(\dfrac{4}{2} \right)^2=(2)^2=4[/tex],
[tex]c_y=\left(\dfrac{-6}{2} \right)^2=(-3)^2=9[/tex].
We add the 4 and 9 to both sides of the equation so that they the equation still makes logical sense.
[tex]x^2+4x+4+y^2-6y+9=3+4+9[/tex]
Now we factor!
[tex](x+2)(x+2)+(y-3)(y-3)=13[/tex]
[tex](x+2)^2+(y-3)^2=13[/tex]
So, our circle has a center as (-2, 3) and a radius of [tex]\sqrt{13}[/tex]!
[tex]\dotfill[/tex]
Graphing
Start by making a dot at the center of the circle, then draw four dots that are about 3.6 units
- above
- below
- left
- right
of the center.
Lastly, draw curved lines that connect those four dot together, forming the circle.
The image below is used by Geogebra.
