# Video: Finding the Equation of a Locus

The figure shows a locus of a point 𝑧 in the complex plane. Write an equation for the locus in the form arg((𝑧 − 𝑎)/(𝑧 − 𝑏)) = 𝜃, where 𝑎, 𝑏 ∈ ℂ and 0 < 𝜃 ⩽ 𝜋 are constants to be found.

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### Video Transcript

The figure shows a locus of a point 𝑧 in the complex plane. Write an equation for the locus in the form the argument of 𝑧 minus 𝑎 over 𝑧 minus 𝑏 equals 𝜃, where 𝑎 and 𝑏, which are complex numbers, and 𝜃, which is greater than zero and less than or equal to 𝜋, are constants to be found.

Remember, the locus of a point 𝑧 in this form is the arc of a circle which subtends an angle of 𝜃 between the points represented by 𝑧 one and 𝑧 two. We have three conditions on 𝜃. If it’s less than 𝜋 by two, the locus is a major arc. If it’s equal to 𝜋 by two, it’s a semicircle. And if it’s greater than 𝜋 by two, the locus is a minor arc. And, remember, the endpoints are not part of this locus. We can see by looking at the diagram that the locus of our 𝑧 is the major arc of a circle. And this makes sense because 𝜃 is equal to 𝜋 by five radians.

The endpoints of our locus lie at 𝐴 and 𝐵 whose Cartesian coordinates are four, negative three and negative three, one, respectively. These represent the complex numbers four minus three 𝑖 and negative three plus 𝑖. And, remember, this locus is traced in a counterclockwise direction. Since the starting point is that represented in the complex number four minus three 𝑖, we can say that the equation of our locus is the argument of 𝑧 minus four minus three 𝑖 over 𝑧 minus negative three plus 𝑖 equals 𝜋 by five. We were actually told to find the value of the constants 𝑎, 𝑏, and 𝜃. 𝑎 is four minus three 𝑖, 𝑏 is negative three plus 𝑖, and 𝜃 is 𝜋 by five.