x^{2}/a^2- y^{2}/b^2= 0 is a "degenerate" conic section. It is the limiting case of then hyperbola x^{2}/a^{2}c- y^{2}/a^{2}c= 1 or x^{2}/a^{2}- y^{2}/b^{2}= c as c goes to 0 and, since it can be factored as (x/a- y/b)(x/a+ y/b)= 0, its graph is the two lines x/a- y/b= 0 and x/a+ y/b= 0.
The eccentricy of such a hyperbola is [tex]\sqrt{ca^2- cb^2}{ca}= \sqrt{a^2- b^2}{\sqrt{c}a}[/tex]. As c goes to 0 that goes to 0. Strictly speaking the eccentricity of a degenerate hyperbola is "not defined" but roughly speaking it is infinity.