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This is true for any conic and pair of straight lines. What about other conics? I.e, if you partially differentiate the equations for a parabola, ellipse, circle, what exactly do you get?

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This is true for any conic and pair of straight lines. What about other conics? I.e, if you partially differentiate the equations for a parabola, ellipse, circle, what exactly do you get?

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radou

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Investigate this link (and click 'next', too): http://www.maths.abdn.ac.uk/~igc/tch/ma1002/diff/node48.html".

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Hurkyl

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It might help to know that "asymptote to a curve" is essentially the same thing as "tangent line to a curve at some point at infinity".

You can shift your perspective (so that the points at infinity become ordinary points) by using a transformation, such as

(s, t) = (y/x, 1/x).

How does your hyperbola, and its asymptotes, look in (s, t)-coordinates?

Oh, if it helps, the reverse transformation is:

(x, y) = (1/t, s/t)

You can shift your perspective (so that the points at infinity become ordinary points) by using a transformation, such as

(s, t) = (y/x, 1/x).

How does your hyperbola, and its asymptotes, look in (s, t)-coordinates?

Oh, if it helps, the reverse transformation is:

(x, y) = (1/t, s/t)

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HallsofIvy

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It's not exactly clear what you mean. The equation of a hyperbola in "standard position" is

This is true for any conic and pair of straight lines. What about other conics? I.e, if you partially differentiate the equations for a parabola, ellipse, circle, what exactly do you get?

[tex]\frac{x^2}{a^2}- \frac{y^2}{b^2}= 1[/itex]

If you differentiate that with respect to x and y you get, respectively,

[tex]\frac{2x}{a}= 0[/tex] and

[tex]\frac{2y}{b}= 0[/tex]

or x= 0, y= 0, certainly NOT the equations of its asymptotes, which are

y= (b/a)x and y= -(b/a)x.

The asymptotes of a hyperbola are the lines the hyperbola tends to for large x and y. One way of finding them is to say that, for very large x, y, "1" is very small and can be neglected in comparison to the other terms: the curve will satisy, approximately, for large x, y

[tex]\frac{x^2}{a^2}- \frac{y^2}{b^2}= 0[/tex]

[tex]\frac{x^2}{a^2}= \frac{y^2}{b^2}[/tex]

[tex]\frac{x}{a}= \pm \frac{y}{b}[/itex]

the equations of the asymptotes.

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