Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 11 / c / i

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- Solution i

Solution

Write

z = x + i y

. Squaring the nonnegative complex modulus on each side gives

(x - 4)^{2} + y^{2} = 4 [(x - 1)^{2} + y^{2}] .

The linear terms cancel, leaving

3 (x^{2} + y^{2}) = 12

. Therefore the locus is

∣ z ∣ = 2, the circle of radius 2 centred at the origin .

Its real and imaginary axis intersections are

\pm 2

and

\pm 2 i

.

Figure 1.
The circular complex locus and an anticlockwise tangential velocity at a general point
. Details.

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