Math Challenges

Submissions for Problem #10

\[ \text{Taking }z=x+iy\text{ show that the hyperbola }x^2-y^2=1\text{ can be expressed as }z^2+\bar{z}^2=2 \]

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Solution: \( \displaylines{\left(x+iy\right)^2+\left(x-iy\right)^2=2\\ x^2+2ixy-y^2+x^2-2ixy-y^2=2\\ 2x^2-2y^2=2\\ x^2-y^2=1} \)