40 Given the parabola $y=x^{2}$, and a real number $c>0$. If there are 2 chords of length 2 passing through the point $(0, c)$, find the range of values for $c$.
40 Let the line $y-c=k x$ passing through $(0, c)$ intersect the parabola $y=x^{2}$, and the length of the chord obtained is 2. Eliminating $y$ yields
$$
x^{2}-k x-c=0
$$

Thus, the difference between the two roots is
$$
x_{2}-x_{1}=\sqrt{k^{2}+4 c} \text {. }
$$

And the chord length is
$$
2=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}=\sqrt{\left(1+k^{2}\right)\left(x_{2}-x_{1}\right)^{2}}
$$

So
$$
4=\left(1+k^{2}\right)\left(k^{2}+4 c\right)
$$

That is
$$
k^{4}+(4 c+1) k^{2}+4 c-4=0 .
$$

Therefore
$$
(4 c+1)^{2}>4(4 c-4)
$$

Thus
$$
16 c^{2}-8 c+15>0
$$
$c>\frac{5}{4}$ or $c<\frac{3}{4}$.