#ellipse geometry help

82 messages · Page 1 of 1 (latest)

tropic magnet
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Math nerds pls help me out!😭 I’m working through ellipse geometry and came across this:

Is the distance from the co-vertex to the focus of an ellipse always equal to the distance from the center to a vertex?

If yes, is there any formal proof or reference (book, paper, website) that supports this claim? I want to be sure it's not just a coincidence from specific examples. Thanks!

main shardBOT
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graceful ridge
tropic magnet
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yes, one of the endpoints of the minor axis

graceful ridge
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So then "vertex" is an endpoint of the major axis?

tropic magnet
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exactly

graceful ridge
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So the question is, is the distance from the co-vertex to the focus equal to the length of the semimajor axis?

tropic magnet
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yess

graceful ridge
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Hmm.

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So an ellipse is, by definition, the set of all points whose sum of distances from the foci is some constant, right?

tropic magnet
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yup 100% right

graceful ridge
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So let's think about the major axis. The major axis would pass through the foci, right?

tropic magnet
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yesyes

graceful ridge
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And the minor axis would be the major axis's perpendicular bisector.

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And the ellipse has reflexive symmetry across both axes.

tropic magnet
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yupp

graceful ridge
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So, let's consider now the vertices and co-vertices.

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If we consider the vertex, what does that tell us about the constant of the ellipse, the "radius" for lack of a better term?

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We know the "radius" must be the sum of the distances from the vertex to the near focus and from the vertex to the far focus, right?

tropic magnet
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yeah so 2a is the constant?

graceful ridge
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2a?

tropic magnet
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I think thats the sum of the distance of vertex to the two foci

graceful ridge
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What is "a"?

tropic magnet
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it's the distance from the center to the vertex

graceful ridge
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So the semimajor axis.

tropic magnet
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yes

graceful ridge
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Then yes. The ellipse constant is the length of the major axis.

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Now, consider the co-vertex. As the endpoint of the minor axis, it's on the perpendicular bisector of the major axis, right?

tropic magnet
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yup

graceful ridge
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Which is also the perpendicular bisector of the line segment between the foci.

tropic magnet
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indeed

graceful ridge
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And what do we know about the perpendicular bisector of a line segment?

tropic magnet
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i think it passes through its midpoint and is equidistant from the segments two endpoints

graceful ridge
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I mean, you just said "it passes through the midpoint" twice in different words.

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This is a lemma you can try to prove yourself, but a point X is equidistant from points A and B if and only if X is on the perpendicular bisector of line segment AB.

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It comes up a lot in geometry.

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It's why all triangles have circumcircles, and why a chord of a circle is uniquely defined by its midpoint.

tropic magnet
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some part of me understands but mostly not

graceful ridge
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Do you understand the statement of the lemma?

tropic magnet
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nope

graceful ridge
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Okay, so. Three points, A, B, and X.

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And we're saying the length of AX is equal to the length of BX if and only if X is on the perpendicular bisector of AB.

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Do you understand? If not, what are you struggling with?

tropic magnet
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ohh ok I finally understand

graceful ridge
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Okay, so then what does that tell us about the co-vertex?

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@tropic magnet

tropic magnet
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that the distance from the co vertex to each vertex is equal??

graceful ridge
tropic magnet
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uh i guess the sum would be more than the length of the major axis

graceful ridge
tropic magnet
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im not really sure but co vertex to the vertices will form a triangle, and in triangle, the two side lengths added together will be I think always more than the length of the base. so yeah that's why i think the sum of those distances will be greater than the major axis

graceful ridge
tropic magnet
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at the minor axis/perpendicular bisector of the major

graceful ridge
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And where else?

tropic magnet
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uh only there?

graceful ridge
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What defines the co-vertex?

tropic magnet
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an endpoint of the minor axis

graceful ridge
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And where does the minor axis end?

graceful ridge
tropic magnet
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at the co vertices

graceful ridge
tropic magnet
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cuz the ellipse is a closed shaped..?

graceful ridge
tropic magnet
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then it has specific distances

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specific lengths

graceful ridge
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And what does that have to do with the minor axis?

tropic magnet
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dang uhm cuz the minor is within that closed shape, it determines the tallness of that shape, so like that's why the segment should be stop at the endpoints/co vertex.

graceful ridge
tropic magnet
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the perimeter of the ellipse is defined as the set of all points where the sum of the distances to the two foci is equal to the length of the major axis. But then let's say the length of the major axis is 10m and as the rule of ellipse states, the sum of the distance of co vertex(a point) to the two focus will also be 10m. But can it be that the distance of co vertex to focus1 be 6m while co vertex to focus2 be 4m? if there are cases like that so then the answer to my question will be no it won't always be equal, right?

graceful ridge
tropic magnet
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ohh so since the co vertex is on the perpendicular bisector of the line segment of the foci then it means that the distance of co vertex to focus A and focus B is equidistant

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right?

graceful ridge
tropic magnet
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ah I see then the answer to my question is yes

graceful ridge
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And since the distances to the foci are the same, and since the sum of those distances is the length of the major axis...

graceful ridge
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A proof is what makes a statement a mathematical fact.

tropic magnet
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Yess, Thank you very much🙏

tropic magnet
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+close

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# prisma veldt

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