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The vis-viva equation is commonly written like this: $$v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right)$$. For $r=a(1-e)$: $$v = \sqrt{GM\left(\frac{2}{a(1-e)} - \frac{1}{a}\right)} = \sqrt{GM\frac{1}{a}\left(\frac{2}{1-e}-1\right)} = \sqrt{GM\frac{1}{a}\left(\frac{1+e}{1-e}\right)}$$. The derivation of the vis-viva equation is not at all trivial and ...