I was asked to prove that the sum of the squares of conjugate diameters in an ellipse are constant. I have thought about it, and tried to figure it out, but I am not sure how to do this. It sounds like the pythagorean theorem could be of help here, but I just can't figure out how!
Proof: Sum of squares of conjugate diameters in an ellipse are constant.
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Can you please define conjugate diameters
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Thank you. I see why it sounds like I am just asking for answers. Conjugate diameters of an ellipse are two lines: One that is parallel to a tangent line and goes through the center, and another that goes through both the center and the point the tangent line passes through. I have been putting a lot of thought into this, and have not come up with very much. At this point, I think I made some progress though. An ellipse is really just a stretched circle, right? So if a circle is \((r\cos(t),r\sin(t))\) an ellipse is the same, but stretched- meaning instead of both the x and y having r, they have different constants? This means that if conjugate diameters in a circle are \(\frac{\pi}{2}\) apart from each other in a circle, the same is true in an ellipse (meaning you can add \(\frac{\pi}{2}\) to t)?
Dr.Peterson
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One thing we need in order to help is your context, so we can suggest things that you know rather than things you don't know, and things that fit into your course rather than things that don't.
Are you expected to use geometrical theorems, or analytic geometry, or calculus, or what? How is the ellipse being defined? What facts about it are available?
Your description of conjugate diameters, using tangents, suggests you might be using calculus. Is that true?
Analytic geometry. I gave you all the available facts. In order to understand it though calculus might help too.
HallsofIvy
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Yes, parametric equations for an ellipse, with center at the origin and axes along the coordinate axes, are x= a cos(t) and y= b sin(t) with parameter t going from 0 to \(\displaystyle 2\pi\). So dx= -a sin(t)dt and dy= b cos(t) dt. The slope of a tangent line at given t= T is dy/dx= -(a/b) tan(T). A line through the center parallel to that tangent line is y= -(a/b)tan(T)x. A line through the point (a cos(T), b sin(T)) and the center is y= (a/b) cot(T)x.
Now when you say "the sum of squares of conjugate diameters" do you mean the sum of squares of their lengths? The line y= -(a/b) tan(T)x crosses the ellipse where b sin(t)= -(a/b) tan(T)(a cos(t) so \(\displaystyle tan(t)= -\frac{a^2}{b^2}tan(T)\) and the line y= (a/b) cot(T)x crosses the ellipse where b sin(T)= (a/b) cot(T)(a cos(t)) so \(\displaystyle tan(t)= \frac{a^2}{b^2} cot(T)\).
Now when you say "the sum of squares of conjugate diameters" do you mean the sum of squares of their lengths? The line y= -(a/b) tan(T)x crosses the ellipse where b sin(t)= -(a/b) tan(T)(a cos(t) so \(\displaystyle tan(t)= -\frac{a^2}{b^2}tan(T)\) and the line y= (a/b) cot(T)x crosses the ellipse where b sin(T)= (a/b) cot(T)(a cos(t)) so \(\displaystyle tan(t)= \frac{a^2}{b^2} cot(T)\).