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To the right is a diagram of a circle. Two pairs of parallel tangents are shown, which together yield a parallelogram circumscribing the circle. Try dragging the ponts A and B so as to make the parallelogram as small as possible. (Do this now before reading on.) You should have found that the minimum area is achieved when the two diameters are perpendicular, making the circumscribing parallelogram a square. No matter where the diameters are, provided they are perpendicular, the area of this square will clearly be constant (equal to the square of the diameter-length, in fact). Let's see if this, or something comparable, is true in an ellipse. |
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The diagrams below show the parallelograms circumscribed at the endpoints of perpendicular diameters of a circle and of an ellipse.
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The above result probably doesn't surprise you at this stage. After all, we've already seen, when looking at the idea of diameters bisecting chords, that perpendicular diameters in an ellipse don't behave like perpendicular diameters in a circle. However, we did discover on that occasion that conjugate diameters in an ellipse behaved like perpendicular chords in a circle. Hence conjugate diameters might be a good bet here too. Let's have a look:
Parallelogram circumscribed at ends of conjugate diameters has constant area
Exercise: Yes, you've guessed it: Prove the above result, (i.e. that the parallelogram circumscribed at endpoints of conjugate diameters in an ellipse has constant area), using, of course, a transformation to the unit circle.
Well, that's the bulk of the course done at this stage. You could get asked other results like these, (the syllabus says "such as...") but in all cases, the idea is to map the situation to the circle, prove whatever you need to prove there, and transfer the result back, making sure to mention what invariances you're relying on.
The only thing left to do is harmonic conjugates leading to the idea of pole and polar. This more or less stands alone from the other material, although it does involve affine invariants, so we get some reults for the ellipse too.