Question

Write the equation of the circle described. a. Center at the origin, containing the point (-6, -8) b. Center (7, 5), containing the point (3, -2).

Transformations of functions
ANSWERED
asked 2021-06-26
Write the equation of the circle described.
a. Center at the origin, containing the point (-6, -8)
b. Center (7, 5), containing the point (3, -2).

Answers (1)

2021-06-27

Use the standard equation of a circle with center (h,k) and radius r:
\(\displaystyle{\left({x}−{h}\right)}^{{2}}+{\left({y}−{k}\right)}^{{2}}={r}^{{2}}\)
a.Using \((h,k)=(0,0)\) and \((x,y)=(−6,−8)\), solve for \(\displaystyle{r}^{{2}}\):
\(\displaystyle{\left(−{6}−{0}\right)}^{{2}}+{\left(−{8}−{0}\right)}^{{2}}={r}^{{2}}\)
\(\displaystyle{36}+{64}={r}^{{2}}\)
\(\displaystyle{100}={r}^{{2}}\)
So, the equation of the circle is:
\(\displaystyle{\left({x}−{0}\right)}^{{2}}+{\left({y}−{0}\right)}^{{2}}={100}\)
\(\displaystyle{x}^{{2}}+{y}^{{2}}={100}\)
b.Using \((h,k)=(7,5)\) and \((x,y)=(3,−2)\), solve for \(\displaystyle{r}^{{2}}\):
\(\displaystyle{\left({3}−{7}\right)}^{{2}}+{\left(−{2}−{5}\right)}^{{2}}={r}^{{2}}\)
\(\displaystyle{16}+{49}={r}^{{2}}\)
\(\displaystyle{65}={r}^{{2}}\)
So, the equation of the circle is:
\(\displaystyle{\left({x}−{7}\right)}^{{2}}+{\left({y}−{5}\right)}^{{2}}={65}\)

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