Circle problem

**mathlete** · March 27th 2008, 11:08 AM

find a formula for the following function

the top half of a circle with center (a,b) and radius r

need some help with this one thanks...

**iknowone** · March 27th 2008, 11:23 AM

Every point $(x,y)$ on the graph of a circle of radius $r$ satisfies the following relationship:

$x^2+y^2 = r^2$

So to express $y$ as a function of $x$ rewrite:

$y = \pm \sqrt{r^2-x^2}$

The top half of the circle is just $y = \sqrt{r^2-x^2}$

Now to shift this function to be centered at $(a,b)$ we have to shift the function up by $b$ and over by $a$ .

$y = \sqrt{r^2-(x-a)^2}+b$

**TheEmptySet** · March 27th 2008, 11:24 AM

Originally Posted by mathlete

find a formula for the following function

the top half of a circle with center (a,b) and radius r

need some help with this one thanks...

The equation of a circle with center (a,b) is given by
$(x-a)^2+(y-b)^2=r^2$

to find the top half we need to isolate y and take the positive square root

$(y-b)^2=r^2-(x-a)^2$ taking the postive root

$y-b=\sqrt{r^2-(x-a)^2 }$

so we get...

$y=b+\sqrt{r^2-(x-a)^2 }$

**topher0805** · March 27th 2008, 11:32 AM

The equation of a circle is:

$(x-h)^2 + (y-k)^2 = r^2$

Where (h,k) is the center of the circle.

The equation for the two halves of a circle are given by isolating y.

$y = \pm \sqrt {r^2 - (x-h)^2} + k$

We are only interested in the top half so we only take the positive square root:

$y = \sqrt {r^2 - (x-h)^2} + k$

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