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Two sirens stand at A(1, 0, 2) and B(5, 4, -2), with all coordinates in metres…

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

Two sirens stand at $A(1, 0, 2)$ and $B(5, 4, -2)$ , with all coordinates in metres. A drone flies where the two sirens are equally loud, which means that it is the same distance from both. Its position $(x, y, z)$ satisfies a plane equation of the form $x + y - z = k$ . Find $k$ . Then find the $z$ -coordinate of the point of this plane that has $x = 4$ and $y = 2$ .

Need a reminder? The key ideaShow

For $PA = PB$ , square both sides. The squares of $x$ , $y$ and $z$ cancel and a plane is left. A fixed ratio $PA = \lambda PB$ ( $\lambda \ne 1$ ) gives a sphere. A point equally far from four points that are not in one plane comes from three linear equations.

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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