the hyperbola H : x2/a2 - y2/b2 = 1, a, b > 0 Since the equation of H, like that of E, involves only square terms in x and y, the symmetry group of H is also {e,h,v,r}. Once again, the existence of reflection symmetries allows us to obtain the sketch from a knowledge of the part of the curve in the first quadrant. Note that H cuts the x-axis at (a,0) and (-a,0), but does not cut the y-axis since we cannot have a real y with - y2/b2 = 1 For (x,y)εH, x2/a2 = 1+ y2/b2, so we have x2 ≥ a2, and hence x ≥ a or x ≤ -a. Thus, H has no points between the lines x = -a and x = a. Some plotting, and application of symmetry will soon produce a picture like that on the right. The hyperbola is shown in blue, the axes of symmetry in green. The lines x = a and x = -a are shown in red. They are the tangents to H at (a,0) and (-a,0). On H, we also have y2/b2 = x2/a2 - 1, so that, as x becomes numerically large, so does y. In fact, we can quantify this in a rather nice way. Suppose that P(x,y) is a point on H in the first quadrant. We can factorize the right-hand side of the equation of E, and rewrite the equation as (x/a + y/b)(x/a - y/b) = 1. Now let x tend to ∞. As we are in the first quadrant, y>0, so y also tends to ∞. Thus, (x/a+y/b) tends to ∞. It then follows that the other factor (x/a-y/b) tends to 0, and, since it positive, does so from above. This shows that the point P will approach the line x/a-y/b = 0, and will do so from below. Applying reflections, we see that the hyperbola H will approach the lines x/a - y/b = 0 and x/a + y/b = 0. These lines are called the asymptotes of H. They are shown in red in the lower picture. some further notation The point O, the centre of the rotation symmetry, is called the centre of the hyperbola. The x-axis is the axis of symmetry which cuts H This is the transverse axis of the hyperbola. The other axis of symmetry, here the y-axis, is called conjugate axis of the hyperbola.
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