Can't Simplify this equation for a Ellipse(Complex Numbers)

Question

I'm asked to sketch the set $\{z \in C : |z + i| + |z + 1| = 2\}$.

I've gotten to the point where I've got the modulus form of $|z + i| + |z + 1|$: $$\sqrt{x^2+(y+1)^2} + \sqrt{(x+1)^2+y^2} = 2$$ How do I change this to the equation for an ellipse?

Answer 1

Let's expand your formula. \begin{align*} &\sqrt{x^2+(y+1)^2}+\sqrt{(x+1)^2+y^2}=2\\ &\iff\sqrt{x^2+(y+1)^2}=2-\sqrt{(x+1)^2+y^2}\\ &\iff x^2+y^2+2y+1=4-4\sqrt{x^2+2x+1+y^2}+x^2+2x+1+y^2\\ &\iff \sqrt{x^2+2x+1+y^2}=1-\frac{1}{2}y+\frac{1}{2}x\\ &\iff \frac{3}{4}x^2+\frac{3}{4}y^2+\frac{1}{2}xy+x+y=0\\ &\iff 3x^2+2xy+3y^2+4x+4y=0 \end{align*}

Note that the last equation represents a conic section. Its discriminant is clearly smaller than 0, so it represents a circle or ellipse. However, it can't be a circle, since we have nonzero $xy$ term.

Actually, we can perform the following change of variables, \begin{equation*} \left\{\begin{aligned} x&=\frac{1}{\sqrt{2}}(u-v)\\ y&=\frac{1}{\sqrt{2}}(u+v)\\ \end{aligned}\right. \end{equation*}

Then, we have \begin{equation*} \frac{(u+\frac{1}{\sqrt{2}})^2}{\frac{1}{2}}+v^2=1 \end{equation*} This is clearly an ellipse