One way:

Solve for y:

y = (1/a)*[a (+/-) 2*sqrt{(x + 1)^2 + a^2}]

For large x this turns into:

y = (1/a)*[a (+/-) 2x] = (+/-)(x/a) + 1 = (+/-)(2x/a)

The asymptotes are (for large x) y = (+/-)x

Thus a = 2.

Below is the graph of (x + 1)^2/4 - (y - 1)^2/4 = -1

-Dan

PS In case anyone saw this whilst I edited it a zillion times, this is my final answer! (And I'm stickin' to it!)