Is -x^2 + 3y^2=3 the equation of the hyperbola with foci is at (2,0) and (-2,0) along with an eccentricity of 2?

Question

This is a problem from a math topic called conic sections. It's a math olympiad question.

Answer 1

Rewrite this in standard form:

-x²/3+y²=1 or x²/3-y²=-1.

Although this is standard form, it represents a hyperbola in which the foci are arranged vertically rather than horizontally. The centre of the hyperbola is at the origin (0,0). The foci lie on the line x=0 which is the y axis.

If a and b are the axes, we have -x²/a²+y²/b²=1. a=√3 and b=1 are related by the eccentricity e, which, for a hyperbola, is greater than 1:

a²=b²(e²-1), that is, 3=e²-1, e²=4, e=2. The focus is at (±be,0)=(2,0) and (-2,0).

So the facts as stated in the question are correct.