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!
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 ...?
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...
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.
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
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 (note that , 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 .