Question

Which of the following is the equation of a circle whose center is at the origin and whose radius is√5 ?

A.) x(squared) + y(squared) =√5

B.) x(squared) + y(squared) = 5

C.) x(squared) + y(squared) = 10

D.) x(squared) + y(squared) = 25

Answer 1

\(x^2 + y^2 = r^2\)
is the equation of a circle with center at the origin and radius r.

Answer 2

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.







Here are some \(\normalsize\color{royalblue}{ \rm examples }\)

\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid royalblue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)

( 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".
\(\normalsize\color{black}{ (x-\color{red}{3})^2+(y-\color{blue}{4})^2=\color{green}{2}^2 }\)
now, simplify \(\normalsize\color{teal}{ ( }\)and this is going to be the \(\normalsize\color{teal}{ \rm ANSWER }\)\(\normalsize\color{teal}{ ) }\)
\(\normalsize\color{teal}{ (x-3)^2+(y-4)^2=4 }\)

\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid royalblue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)

( 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".
\(\normalsize\color{black}{ (x-\color{red}{{\tiny ~}-4})^2+(y-\color{blue}{1})^2=\color{green}{3}^2 }\)
now, simplify \(\normalsize\color{teal}{ ( }\)and this is going to be the \(\normalsize\color{teal}{ \rm ANSWER }\)\(\normalsize\color{teal}{ ) }\)
\(\normalsize\color{teal}{ (x+4)^2+(y-1)^2=9 }\)

\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid royalblue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)

( 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".
\(\normalsize\color{black}{ (x-\color{red}{2})^2+(y-{\tiny ~}\color{blue}{-6})^2=\color{green}{5}^2 }\)
now, simplify \(\normalsize\color{teal}{ ( }\)and this is going to be the \(\normalsize\color{teal}{ \rm ANSWER }\)\(\normalsize\color{teal}{ ) }\)
\(\normalsize\color{teal}{ (x-2)^2+(y+6)^2=25 }\)

\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid royalblue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)

( 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".
\(\normalsize\color{black}{ (x-{\tiny ~}\color{red}{-8})^2+(y-{\tiny ~}\color{blue}{-3})^2=\color{green}{1}^2 }\)
now, simplify \(\normalsize\color{teal}{ ( }\)and this is going to be the \(\normalsize\color{teal}{ \rm ANSWER }\)\(\normalsize\color{teal}{ ) }\)
\(\normalsize\color{teal}{ (x+8)^2+(y+3)^2=1 }\)

\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, brown ,border:2px solid brown ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)

So you can see what the equation of the circle centered each quadrant will look like, with any radius, having the center be (+,+) (-,+) (+,-) or (-,-).

Answer 3

I don't understand