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Find the equation of the tangents of slope -1 to the circle |z - (3 + 3i)| = sqrt(2).

1. Im((z - 4 - 4i)/(-1 + i)) = 0
2. Im((z - 2 - 2i)/(-1 + i)) = 0
3. Re((z - 4 - 4i)/(-1 + i)) = 0
4. Re((z - 2 - 2i)/(-1 + i)) = 0

Correct answer: Im((z - 4 - 4i)/(-1 + i)) = 0

Solution

The circle has centre 3+3i, radius sqrt(2). Tangents of slope -1 touch at 4+4i and 2+2i. Writing the line through 4+4i with direction (-1+i) (slope -1) as Im((z-(4+4i))/(-1+i)) = 0 selects the point 4+4i version matching the options.

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