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 𝑟 is a circle centered at 𝑧 one with a
radius of 𝑟.
Now, comparing our equation to this
one, we can see that we can let 𝑧 one be equal to two and 𝑟 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
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 𝑎, 𝑏 with a radius of 𝑟 is 𝑥 minus
𝑎 all squared plus 𝑦 minus 𝑏 all squared equals 𝑟 squared. And so, the Cartesian equation of
the locus of 𝑧 is 𝑥 minus two all squared plus 𝑦 minus zero all squared equals
three squared. It follows, of course, that we can
rewrite this somewhat as 𝑥 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 𝑥 minus two squared plus 𝑦 squared equals nine.