I wonder if your professor wasn't just hoping you would think about it more yourself- it's a simple calculation.
For any ellipse, the distance, d, from one focus to the ellipse and back to the other focus is a constant.
For an ellipse with major axis a along the x-axis, minor axis b, equation
\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1[/itex]<br />
the total distance from one focus, to the point (a, 0) to the other focus is d= 2a (that should be obvious- the distance you <b>don't</b> cover, from a focus to (-a, 0), is exactly the distance you cover twice, from the other focus to (a, 0)). Now, taking c to be the distance from the center to a focus (so the foci are at (c, 0) and (-c, 0)), the focal distance, we have, by the Pythagorean theorem, that (d/2)<sup>2</sup>= a<sup>2</sup>= b<sup>2</sup>+ c<sup>2</sup> so that c<sup>2</sup>= a<sup>2</sup>- b<sup>2</sup>.<br />
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The "eccentricity" is defined to be the ratio of focal distance to the length of the major-semiaxis, here that is <br />
e= \frac{\sqrt{a^2- b^2}}{a}= \sqrt{1- \frac{b^2}{a^2}}<br />
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The focal parameter, p, is the distance from the focus to the nearest directrix. For an ellipse, a directrix is a line perpendicular to the major axis such that the ratio of the distance from any point on to the nearest focus to the distance from that point to the nearest directrix is equal to the eccentricity. That is, with x the x coordinate of any point on the directix, <br />
\frac{x-a}{a-c}= e[/itex]<br />
Since c= ae, that is<br />
\frac{x- a}{a- ae}= \frac{x- a}{a(1-e)}= e[/itex] &lt;br /&gt;
so that x- a= ae(1-e) and x= a+ ae(1-e)= a(1+ e- e^2).&lt;br /&gt;
Since x is the distance from the center of the ellipse to the directrix, p, the distance from the focus to the directrix is p= x- c= x- ae= a(1+ e- e^2)- ae= a(1- e^2). From that, a= p/(1- e&lt;sup&gt;2&lt;/sup&gt;) which is equivalent to a&lt;sup&gt;2&lt;/sup&gt;= p&lt;sup&gt;2&lt;/sup&gt;/(1- e&lt;sup&gt;2&lt;/sup&gt;)&lt;sup&gt;2&lt;/sup&gt;, of course.&lt;br /&gt;
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Since b&lt;sup&gt;2&lt;/sup&gt;= a&lt;sup&gt;2&lt;/sup&gt;- c&lt;sup&gt;2&lt;/sup&gt;= a&lt;sup&gt;2&lt;/sup&gt;- a&lt;sup&gt;2&lt;/sup&gt;e&lt;sup&gt;2&lt;/sup&gt;= a&lt;sup&gt;2&lt;/sup&gt;(1- e&lt;sup&gt;2&lt;/sup&gt;), we have b&lt;sup&gt;2&lt;/sup&gt;= (p&lt;sup&gt;2&lt;/sup&gt;/(1- e&lt;sup&gt;2&lt;/sup&gt;)&lt;sup&gt;2&lt;/sup&gt;)(1-e&lt;sup&gt;2&lt;/sup&gt;)= p&lt;sup&gt;2&lt;/sup&gt;/(1- e&lt;sup&gt;2&lt;/sup&gt;).