Video: Locus of Points Equidistant from One Point

Describe the locus of 𝑧 such that |𝑧 βˆ’ 2| = 3 and give its Cartesian equation.

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Video Transcript

Describe the locus of 𝑧 such that the modulus of 𝑧 minus two equals three and give its Cartesian equation.

There are several ways to describe the locus of points by using the modulus. In this case, we know that, for a constant complex number 𝑧 one, the locus of a point 𝑧 which satisfies the equation the modulus of 𝑧 minus 𝑧 one is equal to π‘Ÿ is a circle centered at 𝑧 one with a radius of π‘Ÿ.

Now, comparing our equation to this one, we can see that we can let 𝑧 one be equal to two and π‘Ÿ be equal to three. So, we could say that the locus of 𝑧 is a circle centered at the complex number two with a radius of three. But of course, if we were to plot the complex number two on an Argand diagram, we know it would have Cartesian coordinates two, zero. And so, we can say that the locus of 𝑧 is in fact a circle centered at the point two, zero with a radius of three.

Now, the second part of this question asks us to find its Cartesian equation. And so, we recall that the Cartesian equation for a circle centered at π‘Ž, 𝑏 with a radius of π‘Ÿ is π‘₯ minus π‘Ž all squared plus 𝑦 minus 𝑏 all squared equals π‘Ÿ squared. And so, the Cartesian equation of the locus of 𝑧 is π‘₯ minus two all squared plus 𝑦 minus zero all squared equals three squared. It follows, of course, that we can rewrite this somewhat as π‘₯ minus two squared plus 𝑦 squared equals nine.

And so, we found the locus of 𝑧. It’s a circle centered at two, zero with a radius of three. And we’ve seen its Cartesian equation is π‘₯ minus two squared plus 𝑦 squared equals nine.

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