Ellipse

An ellipse is the curve made of all the points whose distances to two fixed points (the foci) always add up to the same total. If you loop a string around two pins and pull it taut with a pencil, the curve you trace is an ellipse, and the length of the string is exactly that constant sum. The orbit of a planet around the Sun is also a slightly stretched ellipse. Drag the semi-major axis a and semi-minor axis b sliders and check that the sum of the distances from a point P to the two foci always stays equal to 2a.

Definition of an Ellipse

Semi-major axis a3

Semi-minor axis b2

👀 What is an Ellipse?

①Two foci F, F' exist

②For any point P on the ellipse, PF + PF' = 2a (constant)

③The larger a, the more elongated the ellipse

④When b is close to a, it approaches a circle

Drawing with String and Pins

📌 String Construction

①Pin both foci

②Tie a string of length 2a between the pins

③Pulling a pencil tight along the string traces an ellipse

④The string length stays constant = PF + PF' = 2a

Standard Form

Standard Form (Horizontal)

x²a² + y²b² = 1 (a > b > 0)

Foci (±c, 0), c² = a² − b²

Standard Form (Vertical)

x²b² + y²a² = 1 (a > b > 0)

Foci (0, ±c), c² = a² − b²

🔍 Deriving the Formula

①Definition: PF + PF' = 2a

②Apply distance formula for P(x,y), F(c,0), F'(-c,0)

③Simplify to x²/a² + y²/b² = 1 (with b² = a² − c²)

④The variable under the larger denominator gives the major axis direction

Eccentricity and Shape

Eccentricity

e = ca (0 < e < 1)

e→0: circle, e→1: very flat ellipse

🎯 Intuition for Eccentricity

①Eccentricity e = c/a = (focal distance)/(semi-major axis)

②e close to 0: foci near center → near circle

③e close to 1: foci far apart → flat

④Planet orbits are ellipses: Earth e≈0.017 (nearly circular), comets e≈0.99 (very flat)

Work It Out

Example 1

For the ellipse x²/25 + y²/9 = 1, find the coordinates of the two foci and the length of the major axis.

Read off a and b, then find c from c² = a² − b².

a²=25, b²=9 ⇒ c²=25−9=16, c=4

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 whose foci are (±3, 0) and whose sum of distances to the foci is 10.

From the distance sum 2a=10 find a, and read c from the foci.

2a=10 ⇒ a=5, c=3

Compute b² = a² − c² and substitute into the standard form.

b²=25−9=16 ⇒ x²25 + y²16 = 1

▸ x²/25 + y²/16 = 1

Get a first from the definition (sum of distances = 2a); read c directly from the foci, and b² follows.

Summary

Key Relation

a² = b² + c²

semi-major² = semi-minor² + focal distance²

2022 KICE mock exam Math (Geometry) type, adapted

What is the distance between the two foci of the ellipse x²/16 + y²/7 = 1?

①6

②5

③4

④3

⑤8

▸ ① 6

Since a²=16 and b²=7, c² = a² − b² = 16−7 = 9, so c=3.

The distance between the two foci is 2c.

2c = 6

🎯 Exam Points

①Ellipse definition: sum of distances to foci = 2a

②Larger denominator in standard form → major axis direction

③Use c² = a² − b² to find focus coordinates

④Eccentricity e = c/a (0 < e < 1)

⑤If center is (h,k): translate x→(x−h), y→(y−k)

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