Question 200474
I'll do the first one to get you started





{{{x^2+y^2+10x+14y+73=0}}} Start with the given equation



{{{x^2+y^2+10x+14y=-73}}}  Subtract {{{73}}} from both sides



{{{(x^2+10x)+(y^2+14y)=-73}}}  Group like terms.



{{{(x+5)^2-25+(y^2+14y)=-73}}} Complete the square for the x terms



{{{(x+5)^2-25+(y+7)^2-49=-73}}} Complete the square for the y terms



{{{(x+5)^2+(y+7)^2-74=-73}}} Combine like terms



{{{(x+5)^2+(y+7)^2=-73+74}}} Add {{{74}}} to both sides



{{{(x+5)^2+(y+7)^2=1}}} Combine like terms



{{{(x-(-5))^2+(y+7)^2=1}}} Rewrite {{{x+5}}} as {{{x-(-5)}}}



{{{(x-(-5))^2+(y-(-7))^2=1}}} Rewrite {{{y+7}}} as {{{y-(-7)}}}



{{{(x-(-5))^2+(y-(-7))^2=1^2}}} Rewrite 1 as {{{1^2}}}




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Notice how the equation is now in the form {{{(x-h)^2+(y-k)^2=r^2}}}. This means that this conic section is a circle where (h,k) is the center and {{{r}}} is the radius.



In this case, {{{h=-5}}}, {{{k=-7}}}, and {{{r=1}}}



So the circle has these properties:


CENTER: (-5,-7)


Radius: {{{r=1}}} (ie the radius is one unit long)