Question

Hi I’m not sure if I got this answer right for my assignment, please help and thank you

Answer 1

`(x+5)^2+(y+6)^2=9`

Step-by-step explanation:

The given equation for a circle is:

`x^2+10x+y^2+12y+52=0`

Re-arrange the equation:

`x^2+y^2+10x+12y+52=0----i`

The equation of a circle center of coordinates, (a, b) and radius, r is given by:

`x^2+y^2-2ax-2by+a^2+b^2-r^2=0----ii`

Comparing (i) and (ii)

`\begin{gathered} -2a=10 \\ a=-(10)/(2) \\ a=-5 \\ Also, \\ -2b=12 \\ b=-(12)/(2) \\ b=-6 \\ \text{And, a}^2+b^2-r^2=52 \\ (-5)^2+(-6)^2-r^2=52 \\ 25+36-r^2=52 \\ 61-r^2=52 \\ r^2=61-52 \\ r^2=9 \end{gathered}`

The equation of a circle in standard form is given by:

`(x-a)^2+(y-b)^2=r^2`

Substitute a = -5, b = -6 and r² = 9 into the equation of a circle in standard form above:

`\begin{gathered} (x--5)^2+(y--6)^2=9 \\ \Rightarrow(x+5)^2+(y+6)^2=9 \end{gathered}`