question form

Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.

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The standard form of an ellipse centered at the origin is

x^2 / a^2 + y^2 / b^2 = 1.

So the first thing you need to do is put the equation into this form.

If you divide both sides by 36 to get the required 1 on the right-hand side the equation becomes

x^2 / 9 + y^2 / 4 = 1.

This is of the form x^2 / a^2 + y^2 / b^2 = 1, provided a^2 = 9 and b^2 = 4. So we conclude that a = 3 and b = 2.

The vertices of this ellipse occur when x = 0 or y = 0. If x = 0 we get

y^2 / 4 = 1

so that

y^2 = 4

and therefore

y = +- 2.

So the ellipse goes through the points (0, 2) and (0, -2).

If y = 0 we get x^2 / 9 = 1, which by steps similar to those just outlined gives us x = +- 3. So the ellipse goes through the points (3, 0) and (-3, 0).

As shown in the qa and elsewhere (e.g., in my Class Notes) the ellipse therefore fits into a rectangle centered at the origin, extending 3 units to the right and left and 2 units up and down.

The longer axis of the ellipse is along the x axis. We call this the major axis, and it runs from (-3, 0) to (3, 0). The semimajor axes run from the center to these vertices, and therefore have length 3.

The minor axis runs from (0, -2) to (0, 2). The semiminor axes have length 2.

The foci of the ellipse lie along its major axis, at distance sqrt( | a^2 - b^2 | ) from the center. In this case a = 3 and b = 2 so the foci lie at distasnce sqrt( | 3^2 - 2^2 | ) = sqrt(5) from the center. The foci are therefore

(-sqrt(5), 0) and (sqrt(5), 0).

If you pick any point on the ellipse and draw a broken line from one focus to that point, then to the other focus, the sum of the lengths of those lines is always the same, no matter what point you choose. Both lines also make equal angles with a line tangent to the ellipse at the chosen point.

The picture you want to associate with an ellipse is the rectangle I mentioned earlier, with the ellipse inside and touching the sides of the rectangle at its vertices, with the two foci along the major axis, and with the broken line from one focus to the ellipse to the other focus.

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