HardMathematicsHigh School (11–12)Coordinate Geometry › Circles and Conic SectionsNumerical

Two gates stand at A(0, 0) and B(6, 0) on a plan, with lengths in metres…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Two gates stand at $A(0, 0)$ and $B(6, 0)$ on a plan, with lengths in metres. A guard walks so that she is always twice as far from gate $A$ as from gate $B$ . Her path is a circle. Find the $x$ -coordinate of its centre, its radius, and the smaller of the two $x$ -values where it crosses the $x$ -axis.

Need a reminder? The key ideaShow

$S + \lambda L = 0$ and $S_1 + \lambda S_2 = 0$ give the circles through the meeting points. A circle touching $L = 0$ at $(x_1, y_1)$ is $(x - x_1)^2 + (y - y_1)^2 + \lambda L = 0$ . For a locus, write the rule for $(h, k)$ and replace $(h, k)$ by $(x, y)$ .

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.

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