high school Ellipse

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 the sum of the distances from a point P to the two foci 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

⑤Slider a is 2~4, b is 1~3.5. If b>a the ellipse is vertical (this sketch still uses 2a)

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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Hyperbola

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