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Suppose we take a standard hyperbola, symmetric about the origin. Then the part of hyperbola in the 1st quadrant would be a mirror image of the part in 2nd quadrant with y axis as mirror; and the part in these two quadrants would be a mirror image of the part below x axis with x axis as mirror.

My question is: If we take only the part of the hyperbola in, say, 1st quadrant and draw a tangent to it from the origin, and then take it's reflection into the other quadrants, we could have a hyperbola with 4 tangents from the origin. Why is symmetry not applicable here?

I'm aware that this question has already been answered, but I'm looking for a more geometric proof, as well as a fault in my reasoning.

DrivenMad
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    There is no line that is tangent to the segment of the parabola in Q1 that goes through the origin. There is a line that will be asymptotic that goes through the origin. And there are lines that are tangent but to not go through the origin. – user317176 Nov 16 '23 at 19:56
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    "I'm aware that this question has already been answered" Please link to the question/answer you've seen, so that people don't spend time telling you things you already know. ... I'll go ahead and mention that the "tangents" to a hyperbola from its center are actually its asymptotes. Correspondingly, the points of tangency are "points at infinity"; while it might look like there are four of them —one in each of four directions— there are actually two —one for each asymptote. In an appropriate context, any line is like a circle that loops around through its single point at infinity. – Blue Nov 16 '23 at 20:36
  • @Blue will do so from now, thanks. Also, I infer that we don't count asymptotes as tangents because they meet the curve at infinity, even if they do satisfy the condition for tangency. – DrivenMad Nov 17 '23 at 04:23
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    @StutiGupta: "I infer that we don't count asymptotes as tangents because they meet the curve at infinity" Context matters. Eg, in projective geometry the pts (and line!) at infinity are no more exotic than their finite counterparts, so a hyperbola's asymptotes are tangents. (Fun Fact: The three types of conics —ellipses, hyperbolas, parabolas— reflect different relationships to the line at infinity —no intersection, crossing, or tangent(!). In projective geometry, that line isn't special, so that all conics are considered "the same"!) – Blue Nov 17 '23 at 05:16
  • @Blue that's a new definition of conics to me. Could you provide a link where I can read about it further? Thanks. – DrivenMad Nov 17 '23 at 14:25
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    @StutiGupta: Do a web search for "projective geometry conic"; explore whatever suits your interest/skill levels. Be advised that the field can be a bit weird. :) ... Even in "regular" geometry, pay attention to how the different conics have lots of curiously-similar properties, especially when it comes to tangents and such, strongly hinting that they're not so different after all. The key is to make friends with the points/line at infinity. (continued) – Blue Nov 17 '23 at 14:40
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    @StutiGupta: (continuing) Consider an ellipse w/one focus at the origin an the other moving along the $x$-axis. When that other focus "gets to infinity", the ellipse becomes a parabola; and when that other focus "passes through infinity and comes back towards the origin from the the other end of $x$-axis" (because the $x$-axis is a loop!), the conic becomes a hyperbola. (See this animation for a sense of this.) You might also consider how the "reflection property" adapts throughout this process. There's a nice unity hidden under it all. :) – Blue Nov 17 '23 at 14:53
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    @Blue thanks! That's definitely a new perspective to see conics with. – DrivenMad Nov 18 '23 at 11:54

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