Question

What is the radius length of the circle with equation

Circles
ANSWERED
asked 2021-08-06
What is the radius length of the circle with equation \(\displaystyle{x}^{{2}}+{y}^{{2}}-{x}+{y}-{4}={0}\)?

Expert Answers (1)

2021-08-07
The standard equation of a circle with center (h,k) and radius rr is given by: \(\displaystyle{\left({x}−{h}\right)}^{{2}}+{\left({y}−{k}\right)}^{{2}}={r}^{{2}}\)
So, we write the given in this form.
Isolate the constant and group the x’s and y’s:
\(\displaystyle{\left({x}^{{2}}−{x}\right)}+{\left({y}^{{2}}+{y}\right)}={4}\)
Complete the square:
\(\displaystyle{\left({\left({x}^{{2}}\right)}-{x}+{\left(\frac{{1}}{{2}}\right)}^{{2}}\right)}+{\left({\left({y}^{{2}}+{y}+{1}{\left(\frac{{1}}{{2}}\right)}^{{2}}\right)}={4}+{\left(\frac{{1}}{{2}}\right)}^{{2}}+{\left(\frac{{1}}{{2}}\right)}^{{2}}\right.}\)
\(\displaystyle{\left({x}-{\left(\frac{{1}}{{2}}\right)}\right)}+{\left({y}+{\left(\frac{{1}}{{2}}\right)}\right)}^{{2}}={4}+\frac{{1}}{{4}}+\frac{{1}}{{4}}\)
\(\displaystyle{\left({x}-{\left(\frac{{1}}{{2}}\right)}\right)}^{{2}}+{\left({y}+\frac{{1}}{{2}}\right)}^{{2}}=\frac{{18}}{{4}}\)
Hence, we can solve for the radius rr by writing: \(\displaystyle{r}^{{2}}=\frac{{18}}{{4}}\)
\(\displaystyle{r}=\frac{\sqrt{{18}}}{{2}}\)
\(\displaystyle{r}={3}\frac{\sqrt{{2}}}{{2}}\)
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