anonymous one year ago 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

1. mathstudent55

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

2. SolomonZelman

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 (-,-).

3. anonymous

I don't understand