# Finding the Equations of a Circle provided a point and the radius

So I tried googling the exact question and I never found the solution. This is homework so I really don't want to know the answer but how to arrive at the answer.

The question that was given was "What are the equations of the circles tangent to the x-axis of radius 4 and whose center is at (-2,y)?"

From the given and a little graphing, I found that (-2,0) would be the point where the circle is tangent to the x-axis. Am I correct?

if I am correct, then I have 2 given pieces of information which is the radius of 4 and a point as well as the x-value of the center.

With all the equations my professor given me, I still don't know how to solve to get the center. Can someone point me to the right direction?

thanks.

• Hint: $(x-a)^2+(y-b)^2=r^2$ Is a circle radius $r$ centred at $(a,b)$ – Karl Feb 20 '16 at 13:56
• If the $x$ axis is a tangent then there are 2 possible circles. One of which is centred at $(-2,4)$ what is the other? – Karl Feb 20 '16 at 14:01

Let's use $(a,b)$ to denote the coordinates of the center of the circle.

The circle has radius $4$ and it's tangent to the $x$-axis, so its center must be $4$ units above or below the $x$-axis. This means that $b=4$ or $b=-4$.

You're also told (not very clearly) that the center coordinates have the form $(-2,y)$, for some $y$. This means that $a=-2$.

So, now we know that the center is at $(a,b) = (-2,4)$, or $(a,b) = (-2,-4)$, and we know the radius is $4$, so you can probably figure out the equations of the two circles.

• No problem upvoted – Karl Feb 20 '16 at 14:12

Hint:

You have correctly found the point $(-2,0)$ where the circle is tangent to the $x$ axis. Now you know the radius $4$ and you also know that the radius is orthogonal to the tangent. So, where can be the center of the circle?