Question:
Find the equation of the ellipse, coordinates of whose foci are $(1,0)$ and $(-1,0)$ and the length of whose latera recta is $3$.
My book's attempt:
$$2\sqrt{a^2-b^2}=2$$
$$b^2=a^2-1\tag{1}$$
Now,
$$\frac{2b^2}{a}=3$$
$$\frac{2(a^2-1)}{a}=3$$
$$a=2,-\frac{1}{2}\tag{2}$$
Now immediately after line $(2)$, my book asserted that $-\frac{1}{2}$ isn't an acceptable value of $a$, but how did my book know this just by looking at it? I only understood that $-\frac{1}{2}$ wasn't acceptable by inputting it in $(1)$: we get $b^2=-\frac{3}{4}$, which is impossible.
So, how did my book know that $a\neq-\frac{1}{2}$ just by looking at it?