An ellipse is the curve made of all points whose distances to two fixed points, the foci, add up to the same total.
Tie the ends of a string to two pins, pull it tight with a pencil and go once around: the pencil draws an ellipse, and the length of the string is that constant total.
The orbits of the planets around the Sun are ellipses too, with the Sun at one of the two foci.
Change the horizontal semi-axis a and the vertical semi-axis b: the foci move with them, and the sum of the distances from P to the two foci always equals the length of the long axis.
Definition of an Ellipse
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The two foci of x²/a² + y²/b² = 1 and the distances from a point P on the curve to them
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👀 What to read in the picture
①The two red points are the foci F and F′, and the green segments are the distances from P to them
②However you change a and b, PF′ + PF in the top band equals twice the longer semi-axis
③If a > b, the ellipse is wider than it is tall, the foci lie on the x-axis at (±c, 0), and c² = a² − b²
④If b > a, the ellipse is taller than it is wide, the foci move to the y-axis at (0, ±c) with c² = b² − a², and the sum of the distances is 2b
⑤If a = b, the two foci meet at the center and the ellipse becomes a circle
Drawing with a String and Pins
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Pins at the two foci and a string of fixed length drawing the ellipse
📌 The string method
①Put a pin at each focus
②Tie the ends of a string as long as the long axis to the two pins
③Keep the string taut with a pencil and go once around: the pencil draws an ellipse
④Since the string does not change length, the sum of the distances from the pencil point to the two pins stays constant
Standard Form
Standard form (wider than tall)
x²a² + y²b² = 1 (a > b > 0)
Foci (±c, 0), c² = a² − b²
Standard form (taller than wide)
x²b² + y²a² = 1 (a > b > 0)
Foci (0, ±c), c² = a² − b²; here a is the longer semi-axis, so it sits under y²
🔍 Setting up the equation
①The definition is PF + PF′ = 2a
②Apply the distance formula to P(x, y), F(c, 0) and F′(−c, 0)
③Clearing the square roots one at a time gives x²/a² + y²/b² = 1 with b² = a² − c²
④The variable over the larger denominator gives the direction of the long axis
Eccentricity and Shape
Eccentricity
e = ca (0 < e < 1)
If e is close to 0 the ellipse is nearly a circle; if e is close to 1 it is very flat
🎯 What eccentricity means
①The eccentricity e = c/a is the distance from the center to a focus divided by the longer semi-axis
②When e is near 0, the foci crowd toward the center and the ellipse is nearly a circle
③When e is near 1, the foci spread toward the ends and the ellipse flattens
④Earth's orbit has eccentricity about 0.017, almost a circle, while Halley's Comet has about 0.97, very flat
Worked Examples
Example 1
Find the coordinates of the foci and the length of the major axis of the ellipse x²/25 + y²/9 = 1.
1
Read a and b, then find c from c² = a² − b².
a² = 25, b² = 9 ⇒ c² = 25 − 9 = 16, c = 4
2
The foci are (±c, 0) and the major axis has length 2a.
foci (±4, 0), major axis 2a = 10
▸ Foci (±4, 0), major axis length 10
The larger denominator, 25, is under x², so the major axis is horizontal and the foci lie on the x-axis.
Example 2
Find the equation of the ellipse with foci (±3, 0) whose distances to the foci always add up to 10.
1
From the sum 2a = 10 find a, and read c from the foci.
2a = 10 ⇒ a = 5, c = 3
2
Find b² from b² = a² − c² and substitute into the standard form.
b²=25−9=16 ⇒ x²25 + y²16 = 1
▸ x²/25 + y²/16 = 1
Getting a from the definition (sum of distances = 2a) and c from the foci fixes b².