Three circular fountains have the edges x² + y² + 4x + 2y - 6 = 0, x² + y² - 12x - 2y + 30 = 0 and…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
Three circular fountains have the edges , and , with lengths in metres. A guard stands at the point from which the tangent lengths to the three fountains are equal. Find the coordinates of that point and the length of each tangent.
Need a reminder? The key ideaShow
The radical axis of and is , and the tangent lengths from any point of it are equal. The radical centre of three circles is the point where their three radical axes meet. Circles are orthogonal when .
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. Two radical axes. The first minus the second gives , which is . The first minus the third gives , which is .
Step 2. The meeting point. Adding and gives , so and . This is the radical centre. It also lies on the third radical axis, because is , which is at .
Step 3. The tangent length. . So the length is metres. The other two circles give and .
Answer: the point is and each tangent is metres long.
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