A moving circular disc of radius 2 centimetres touches two fixed discs from outside…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A moving circular disc of radius centimetres touches two fixed discs from outside. The fixed discs have the edges and , with lengths in centimetres. Find the coordinates of the centre of the moving disc in the upper half of the plane, to two decimal places.
Need a reminder? The key ideaShow
and give the circles through the meeting points. A circle touching at is . For a locus, write the rule for and replace by .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The two distances. The first fixed disc has centre and radius , so the moving centre is centimetres from it. The second has centre and radius , so the moving centre is centimetres from it.
Step 2. Two equations. and .
Step 3. Subtract. , which is , so .
Step 4. The height. , so centimetres in the upper half.
Answer: about . The centre always satisfies centimetres, which is why it moves along a branch of a hyperbola.
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