Why does the x value and p value of asymptotes in a hyperbola have different signs?

Question

Why is the x value opposite to the p value in sign on a hyperbola

Answer 1

Standard equation of a hyperbola is:

x²/a²-y²/b²=1 or x²/a²-y²/b²=-1 and the equations of its asymptotes are found from:

x²/a²-y²/b²=0, b²x²=a²y², that is, bx=ay and bx=-ay. These lines can be written:

y=bx/a and y=-bx/a.

If the centre (origin) of the hyperbola is (h,k) then the standard equation becomes:

(x-h)²/a²-(y-k)²/b²=±1 with corresponding adjustments to the equations of the asymptotes.

Please define what you mean by p. My guess is that p=b/a or p=-b/a, being the slope of the two asymptote lines. The sign simply means that one asymptote slopes to the right and the other slopes to the left, so that the asymptotes cross one another at the centre of the hyperbola. In a rectangular hyperbola the asymptotes are each inclined at 45° to the x and y axes and are perpendicular to each other (they cross at right angles).