Answer: The co-ordinates of the center of the circle are (-7, -1) and the length of the radius is 6 units.
Step-by-step explanation: Given that a circle is described by the following equation :
[tex]x^2+y^2+14x+2y+14=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We are to find the co-ordinates for the center and the length of the radius of the given circle.
The STANDARD equation of a circle with center (h, k) and radius of length r units is given by
[tex](x-h)^2+(y-k)^2=r^2.[/tex]
From equation (i), we have
[tex]x^2+y^2+14x+2y+14=0\\\\\Rightarrow (x^2+14x)+(y^2+2y)+14=0\\\\\Rightarrow (x^2+2\times x\times 7+7^2)+(y^2+2\times y\times1+1^2)+14-7^2-1^2=0\\\\\Rightarrow (x+7)^2+(y+1)^2+14-49-1=0\\\\\Rightarrow (x+7)^2+(y+1)^2-36=0\\\\\Rightarrow (x+7)^2+(y+1)^2=36\\\\\Rightarrow (x-(-7))^2+(y-(-1))^2=6^2.[/tex]
Comparing with the standard form, we get
center, (h, k) = (-7, -1) and radius, r = 6 units.
Thus, the co-ordinates of the center of the circle are (-7, -1) and the length of the radius is 6 units.
