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Math Help - Complex Numbers (Loci)

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    Complex Numbers (Loci)

    Hello!

    Q: The complex number w is represented by the point of intersection of the loci |z -8 -6i| = 5 and arg(z -4 -3i) = 2 \alpha, where \alpha = \arctan (0.75). Find w, in the form x + iy, giving the exact decimal values of x and y.

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    Quote Originally Posted by Tangera View Post
    Hello!

    Q: The complex number w is represented by the point of intersection of the loci |z -8 -6i| = 5 and arg(z -4 -3i) = 2 \alpha, where \alpha = \arctan (0.75). Find w, in the form x + iy, giving the exact decimal values of x and y.

    Thank you for helping!
    There is probably some simple geometric way of doing this - it can't be coincidence that the ray \text{arg} (z - 4 -3i) = \alpha passes through the centre of the circle |z -8 -6i| = 5 and that the terminus of the ray lies on this circle.

    But a blunt apporach is:

    Get the Cartesian equation of |z -8 -6i| = 5, easy if you recognise it as a circle of radius 5 and centre at (8, 6).

    Get the equation of the line that the ray \text{arg} (z - 4 -3i) = 2 \alpha lies on. Note that the gradient of this line is \tan 2 \alpha = \frac{2 \tan \alpha}{1 - \tan^2 \alpha} and it passes through the point (4, 3).

    Get the intersection of the circle and the line. The solution you want to keep is the one such that y > 3 (a diagram makes this clear).
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