🔮 Conic Sections
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Ellipses

Write and interpret the standard equation of an ellipse.

An ellipse is the set of all points where the sum of distances from two fixed foci is constant. The standard equation has two forms depending on orientation of the major axis.

(x−h)2a2+(y−k)2b2=1(a>b>0)\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \qquad (a > b > 0)

Ellipse Components

Center: (h, k)Major axis length: 2a (along the larger denominator)Minor axis length: 2bFoci distance from center: c = √(a² − b²)Sum of focal distances for any point: 2a
c2=a2−b2Eccentricity: e=ca(0<e<1)c^2 = a^2 - b^2 \qquad \text{Eccentricity: } e = \frac{c}{a} \quad (0 < e < 1)
✏️Analyze x²/25 + y²/9 = 1
a² = 25, b² = 9, so a = 5, b = 3. Major axis is horizontal (a² under x²), length 10. Minor axis length 6. c = √(25−9) = √16 = 4. Foci: (±4, 0).
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Remember This!

If a² is under x², the major axis is horizontal (wider left-right). If a² is under y², the major axis is vertical (taller). Always check which denominator is larger.

✏️ Try It!

For the ellipse x²/16 + y²/25 = 1, which axis is the major axis?

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