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Describe the locus of π§ such that the modulus of π§ minus two equals three and give its Cartesian equation.
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There are several ways to describe the locus of points by using the modulus.
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In this case, we know that, for a constant complex number π§ one, the locus of a point π§ which satisfies the equation the modulus of π§ minus π§ one is equal to π is a circle centered at π§ one with a radius of π.
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Now, comparing our equation to this one, we can see that we can let π§ one be equal to two and π be equal to three.
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So, we could say that the locus of π§ is a circle centered at the complex number two with a radius of three.
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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.
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And so, we can say that the locus of π§ is in fact a circle centered at the point two, zero with a radius of three.
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Now, the second part of this question asks us to find its Cartesian equation.
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And so, we recall that the Cartesian equation for a circle centered at π, π with a radius of π is π₯ minus π all squared plus π¦ minus π all squared equals π squared.
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And so, the Cartesian equation of the locus of π§ is π₯ minus two all squared plus π¦ minus zero all squared equals three squared.
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It follows, of course, that we can rewrite this somewhat as π₯ minus two squared plus π¦ squared equals nine.
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And so, we found the locus of π§.
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Itβs a circle centered at two, zero with a radius of three.
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And weβve seen its Cartesian equation is π₯ minus two squared plus π¦ squared equals nine.