Ellipse: Foci, Vertices, and the Constant Sum Property

An ellipse is one of the four conic sections, and it has a beautiful geometric definition: it's the set of all points where the sum of distances to two fixed points (the foci) is constant.

Standard form

For an ellipse centered at the origin with the major axis along the x-axis:

$\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1, \quad a > b$

• $a$ = semi-major axis (half the longest diameter)
• $b$ = semi-minor axis (half the shortest diameter)
• Foci are at $(\pm c, 0)$ where $c = \sqrt{a^{2} - b^{2}}$
• The constant sum = $2a$ (the length of the major axis)

Solving the problem

For $x^{2}/25 + y^{2}/9 = 1$: $a^{2} = 25 \Rightarrow a = 5$, $b^{2} = 9 \Rightarrow b = 3$.

$c = \sqrt{25 - 9} = \sqrt{16} = 4$

• Vertices (endpoints of major axis): $(\pm 5, 0)$
• Co-vertices (endpoints of minor axis): $(0, \pm 3)$
• Foci: $(\pm 4, 0)$
• Constant sum: $2a = 10$

Verification: Take the point $(5, 0)$ (a vertex):

• $PF_{1} = |5 - 4| = 1$
• $PF_{2} = |5 - (-4)| = 9$
• Sum = $1 + 9 = 10$ ✓

Take $(0, 3)$ (a co-vertex):

• $PF_{1} = \sqrt{16 + 9} = 5$
• $PF_{2} = \sqrt{16 + 9} = 5$
• Sum = $5 + 5 = 10$ ✓

The eccentricity

$e = \dfrac{c}{a} = \dfrac{4}{5} = 0.8$

Eccentricity ranges from 0 (circle) to just below 1 (very elongated). At $e = 0$, the foci merge at the center and the ellipse becomes a circle. Our ellipse at $e = 0.8$ is fairly elongated.

Real-world ellipses

• Planetary orbits: Kepler's first law says planets orbit the sun in ellipses with the sun at one focus. Earth's orbit has $e = 0.017$ (nearly circular); Pluto's has $e = 0.25$.
• Whispering galleries: Elliptical rooms (like the U.S. Capitol's Statuary Hall) focus sound from one focus to the other — whisper at one focus, hear clearly at the other.
• Medical lithotripsy: Shock waves generated at one focus of an ellipsoidal reflector converge at the other focus, shattering kidney stones without surgery.

Try it in the visualization

Drag the $a$ and $b$ sliders to change the ellipse shape. Toggle "constant sum" to see the two distance lines from any point on the ellipse to the foci — their sum is always $2a$. Animate the tracing point and watch the distances stretch and compress while their sum stays constant. Toggle "eccentricity" to see how $e = c/a$ changes as you adjust the shape.