Ellipse definition

Question

Spivak defines an ellipse as the set of points, the sum of whose distances from two fixed points is a constant. He takes these two points to be $(-c,0)$ and $(c,0)$ and the sum of the distances to be $2a$. He then derives $x^2/a^2 + y^2/(a^2-c^2) = 1$ and that 'clearly' we must choose $a>c$. Why must this choice be made?

Answer 1

Spivak defines an ellipse as the set of points, the sum of whose distances from two fixed points is a constant. He takes these two points to be $(-c,0)$ and $(c,0)$ and the sum of the distances to be $2a$.

If $a<c$, then the ellipse does not contain any points. Therefore, we must have $a\geq c$ for the definition to make sense. Further, if $a=c$, then the ellipse is the line segment from $(-c,0)$ to $(c,0)$, which may or may not be what you want. But the case $a=c$ can't be expressed by the equation, because the denominator $a^2-c^2$ would be zero.

Answer 2

Well if

$a < c$

$a^2 < c^2$

$a^2-c^2 < 0 $

Let $a^2-c^2 = -d^2$

Equation would then be:

$x^2/a^2 - y^2/d^2 = 1$

A hyperbola