Cartesian Form

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We can easily convert this into Cartesian form by replacing z by x+iy
|(x+iy)-(-3 +2i)|=5
|(x+3) + i(y-2)|=5
√(〖(x+3)〗^2+〖(y-2)〗^2 ) =5 or 〖(x+3)〗^2+〖(y-2)〗^2=25
Which is the exact form of a Cartesian equation of a circle with a center of (-3, 2) and a radius of 5
We could check if a particular point lies on the circle Ex. z2=1-3i
Substituting into the equation of the circle, with z replaced by z2=1-3i
|(1-3i) - (-3 + 2i)|=5
|4-5i|=5
√(4^2+〖(-5)〗^2 ) =5
41=5, which is not true, thus z2 does not lie on the circle This procedure, of course, is a little longer than substituting a point in the Cartesian form and checking. There are more steps involved here in simplifying the complex number and also finding the modulus.

2. To find the equation of a circle with a given center and
…show more content…
Of course finding the center and the radius is a complicated procedure.

4. Equation of the Arc of a Circle In all the previous cases, we looked at complete circles. I was curious to find out if parts of circles, like an arc, could be represented using complex numbers. I have never done this in Math SL, so I was curious in looking at this possibility. Interestingly, I discovered that the arc of a circle can be considered as the locus of a point, which moves such that the angle formed at that point, by two other given points, is a constant. This is a concept that we study in earlier grades. In Cartesian form, it is difficult to represent this arc by an equation. All that can be done is to write the equation of the complete circle and provide some restrictions for x and y. But I found that this can be conveniently represented by a complex equation. For this I will have to bring in another concept in complex numbers, which is the angle between two lines.

Angle between two line

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