Definition 5.8.
An ellipse is the set of points in a plane the sum of whose distances from two fixed points \(F_1\) and \(F_2\) (the foci) is constant.
Section 5.2 Ellipse
For \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) with \(a \gt b\text{,}\) the segment joining \((a,0)\) and \((-a,0)\) is the semi-major axis, the segment joining \((0,b)\) and \((0,-b)\) is the semi-minor axis, and the foci satisfy \(c^2 = a^2 - b^2 \gt 0\text{.}\)
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Where do the formulas for \(a\) come from? Two facts do all the work. First, evaluating Definition 5.8 at the vertex \((a,0)\) shows that the two focal radii there are \(a-c\) and \(a+c\text{,}\) so their sum is
\begin{equation*}
r_1 + r_2 = (a-c) + (a+c) = 2a,
\end{equation*}
which identifies the constant in the definition as \(2a\text{;}\) equivalently \(a = \frac{1}{2}(r_1 + r_2)\text{.}\) Second, evaluating at the co-vertex \((0,b)\) forces \(r_1 = r_2 = a\) by symmetry, and that radius is the hypotenuse of a right triangle with legs \(b\) and \(c\text{.}\) Pythagoras then gives
\begin{equation*}
a^2 = b^2 + c^2 \qquad\Longleftrightarrow\qquad c^2 = a^2 - b^2.
\end{equation*}
So for the ellipse in Figure 5.9, \(a\) is the hypotenuse of that triangle, which is why \(a\) is the largest of the three lengths and why \(c^2 = a^2 - b^2\) carries a minus sign.
Example 5.12 Horizontal ellipse.
Sketch the graph of \(4x^2 + 9y^2 = 36\) and locate the foci.
Solution.
Dividing through by \(36\) puts the equation in standard form:
\begin{equation*}
\frac{x^2}{9} + \frac{y^2}{4} = 1.
\end{equation*}
The larger denominator sits under \(x^2\text{,}\) so the major axis is horizontal with \(a = 3\) and \(b = 2\text{.}\) Then
\begin{equation*}
c^2 = a^2 - b^2 = 9 - 4 = 5 \;\Longrightarrow\; c = \sqrt{5},
\end{equation*}
so the foci are \(F_1(\sqrt{5},0)\) and \(F_2(-\sqrt{5},0)\text{.}\)
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Example 5.15 A vertical ellipse (foci on the \(y\)-axis).
Sketch the graph of \(25x^2 + 16y^2 = 400\) and locate the foci.
Solution.
Dividing by \(400\text{,}\)
\begin{equation*}
\frac{x^2}{16} + \frac{y^2}{25} = 1.
\end{equation*}
This time the larger denominator, \(25\text{,}\) sits under \(y^2\text{,}\) so the major axis is vertical. The semi-major axis is \(a = \sqrt{25} = 5\) (along the \(y\)-axis) and the semi-minor axis is \(b = \sqrt{16} = 4\) (along the \(x\)-axis). Hence
\begin{equation*}
c^2 = a^2 - b^2 = 25 - 16 = 9 \;\Longrightarrow\; c = 3.
\end{equation*}
Because the major axis is vertical, the foci lie on the \(y\)-axis: \(F_1(0,3)\) and \(F_2(0,-3)\text{.}\) The lesson of the twist: read off the larger denominator first, since it decides which axis carries the foci.
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