A curve may be both asymptotic and a line of curvature, in which case the curve is a line (such as the rulings of a ruled surface).

yes, an asymptote is a curve that gets closer but never touches the x axis.

The domain of the Normal distribution is the whole of the real line. As a result the horizontal axis is asymptotic to the Normal distribution curve. The curve gets closer and closer to the axis but never, ever reaches it.

A hyperbola is a conic section. It is a curve in 2-dimensional space whose equation is of the form (x - a)*(y - b) = c where a, b and c are constants.

The curve is asymptotic at the points x = a and y = b.

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There are no intercepts because the curve, xy = 4 is asymptotic. When x = 0 (where the y intercept would be) y is infinite, and conversely, when y = 0 x is infinite.