# Thread: Need help with Geometry and Loci

1. ## Need help with Geometry and Loci

I need help to answer this question... what is the locus of the centers of all circles with a radius of 2 that are tangent to a given line y? This one has me stumped!

2. Originally Posted by lanzcarp
I need help to answer this question... what is the locus of the centers of all circles with a radius of 2 that are tangent to a given line y? This one has me stumped!
If the given line is of the form y = k, then it's not hard to see that the locus consists of two lines, namely y = k - 2 and y = k + 2. However .....

Did you have a particular NON-horizontal line in mind (eg. y = x) or did you want the locus for the general non-horizontal line y = mx + c ...?

3. NO information is given about the line at all.

4. Originally Posted by lanzcarp
NO information is given about the line at all.
Well, you've got the answer when the line is horizontal.

When the line has the form y = mx + c, you should note that the locus of the centre of the circle will be the two lines that are parallel to y = mx + c and at a perpendicular distance of 2 ..... A simple sketch should reveal this to you.

That should get you started. Feel free to post some follow-up questions if you're still stuck.

I'm still confused- I don't see how the center of a circle can be tangent to a line to begin with. Doesn't that contradict the definition of tangent ( a line in the plane of the circle that intersects the circle in exactly one point, called the point of tangency). If the center of the circle is tangent to a line, then the line is touching more than one point in the circle...

6. Originally Posted by lanzcarp
I'm still confused- I don't see how the center of a circle can be tangent to a line to begin with. Doesn't that contradict the definition of tangent ( a line in the plane of the circle that intersects the circle in exactly one point, called the point of tangency). If the center of the circle is tangent to a line, then the line is touching more than one point in the circle...
You misread the question. The center of the circle isn't tangent to the line, the circle is tangent to the line:
What is the locus of the centers of all circles with a radius of 2 that are tangent to a given line y?
-Dan

7. Originally Posted by mr fantastic
Well, you've got the answer when the line is horizontal.

When the line has the form y = mx + c, you should note that the locus of the centre of the circle will be the two lines that are parallel to y = mx + c and at a perpendicular distance of 2 ..... A simple sketch should reveal this to you.

That should get you started. Feel free to post some follow-up questions if you're still stuck.
I guess that if you only have to describe the locus, then there's not much more to ask or say.

But if you want to get the equations of those two locus lines that are parallel to y = mx + c, you might want some further help ....

The easiest way of getting them I think would be to first get the equation of two lines parallel to y = x and at a distance 2 units from x. Then multiply in the dilation factor of m, followed by the vertical translation of c.

8. ## makes sense now

Ok--thanks Dan and Mr. F; our instructors are famous for asking trick questions, which is why I wondered about the center being the point of tangency, so I will assume they mean the circle and not the center. Also, we haven't gone over equations for loci, but I can see the parallel lines that would be the set of loci. Thanks to both of you

9. Originally Posted by lanzcarp
our instructors are famous for asking trick questions, I can see the parallel lines that would be the set of loci.
After seeing this remark and reading other responses I will add one more.
The locus of the centers of all circles with a radius of 2 that are tangent to a given line is the set of points 2 units from that line.
Given the line $\displaystyle Ax + By + C = 0$ (note that $\displaystyle A^2 + B^2 \ne 0$, otherwise the line is degenerate) the locus of the centers of all circles with a radius of 2 that are tangent to that given line is $\displaystyle \left\{ {(p,q):\frac{{\left| {Ap + Bq + C} \right|}}{{\sqrt {A^2 + B^2 } }} = 2} \right\}$.