Question

Find the equation of the circle whose center and radius are given.

center (0, 8), radius = 8

Answer 1

The (vertex form of the) equation of a circle is:
\(\large\color{black}{ \displaystyle (x-{\rm \color{red}{h}} )^2+ (y-{\rm \color{blue}{k}} )^2 = {\rm \color{green}{r}}^2 }\)

Where
1) \(\large\color{black}{ \displaystyle({\rm \color{red}{h}} ,~{\rm \color{blue}{k}} )}\) is the \(\normalsize\color{red}{ \bf c\color{blue}{e}n\color{blue}{t}e\color{blue}{r} }\) of the circle.
2) \(\large\color{black}{ \displaystyle {\rm \color{green}{r}} }\) is the \(\normalsize\color{green}{ \bf radius }\) of the circle.


\(\normalsize\color{royalblue}{ \rm I~posted~more~examples~below~{\bf \dots} }\)

\(\scriptsize\color{#003399 }{\scriptsize{\bbox[5pt, #003399 ,border:2px solid #003399 ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)\({\tiny\\[0.7em]}\)
( 1 ) QUESTION:
`Find the equation of a circle with a center (3,4) and radius 2.`

( 1 ) SOLUTION:
\(\normalsize\color{black}{ (x-h)^2+(y-k)^2=r^2 }\)
plug in your "h" and "k".
\(\normalsize\color{black}{ (x-\color{red}{3})^2+(y-\color{blue}{4})^2=\color{green}{2}^2 }\)\({\tiny\\[0.7em]}\)
now, simplify \(\normalsize\color{#996600}{ ( }\)and this is going to be the \(\normalsize\color{#996600}{ \rm ANSWER }\)\(\normalsize\color{#996600}{ ) }\)
\(\normalsize\color{#996600}{ (x-3)^2+(y-4)^2=4 }\)

\(\scriptsize\color{#003399 }{\scriptsize{\bbox[5pt, #003399 ,border:2px solid #003399 ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)\({\tiny\\[0.7em]}\)
( 2 ) QUESTION:
`Find the equation of a circle with a center (-4,1) and radius 3.`

( 2 ) SOLUTION:
\(\normalsize\color{black}{ (x-h)^2+(y-k)^2=r^2 }\)
plug in your "h" and "k".
\(\normalsize\color{black}{ (x-\color{red}{{\tiny ~}-4})^2+(y-\color{blue}{1})^2=\color{green}{3}^2 }\)\({\tiny\\[0.7em]}\)
Now, simplify \(\normalsize\color{#996600}{ ( }\)and this is going to be the \(\normalsize\color{#996600}{ \rm ANSWER }\)\(\normalsize\color{#996600}{ ) }\)
\(\normalsize\color{#996600}{ (x+4)^2+(y-1)^2=9 }\)

\(\scriptsize\color{#003399 }{\scriptsize{\bbox[5pt, #003399 ,border:2px solid #003399 ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)\({\tiny\\[0.7em]}\)
( 3 ) QUESTION:
`Find the equation of a circle with a center (2,-6) and radius 5.`

( 3 ) SOLUTION:
\(\normalsize\color{black}{ (x-h)^2+(y-k)^2=r^2 }\)
plug in your "h" and "k".
\(\normalsize\color{black}{ (x-\color{red}{2})^2+(y-{\tiny ~}\color{blue}{-6})^2=\color{green}{5}^2 }\)\({\tiny\\[0.7em]}\)
Now, simplify \(\normalsize\color{#996600}{ ( }\)and this is going to be the \(\normalsize\color{#996600}{ \rm ANSWER }\)\(\normalsize\color{#996600}{ ) }\)
\(\normalsize\color{#996600}{ (x-2)^2+(y+6)^2=25 }\)

\(\scriptsize\color{#003399 }{\scriptsize{\bbox[5pt, #003399 ,border:2px solid #003399 ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)\({\tiny\\[0.7em]}\)
( 4 ) QUESTION:
`Find the equation of a circle with a center (-8,-3) and radius 1.`

( 4 ) SOLUTION:
\(\normalsize\color{black}{ (x-h)^2+(y-k)^2=r^2 }\)
plug in your "h" and "k".
\(\normalsize\color{black}{ (x-{\tiny ~}\color{red}{-8})^2+(y-{\tiny ~}\color{blue}{-3})^2=\color{green}{1}^2 }\)\({\tiny\\[0.7em]}\)
Now, simplify \(\normalsize\color{#996600}{ ( }\)and this is going to be the \(\normalsize\color{#996600}{ \rm ANSWER }\)\(\normalsize\color{#996600}{ ) }\)
\(\normalsize\color{#996600}{ (x+8)^2+(y+3)^2=1 }\)
\({\tiny\\[0.8em]}\)
I hope that this helps you out ....