Key takeaway: An ellipse is the oval made of all points whose distances to two fixed foci add up to one constant, centered at the origin when b²x² + a²y² = a²b².
Grade 12 – PrecalculusG-GPE.A.3Geometry and Measurement
Definition of Ellipse
An ellipse is the oval you get when you pin both ends of a loose string at two points, pull the string tight with a pencil, and trace around. The two pins are the foci, and the string length stays fixed, so the sum of the pencil’s distances to the pins never changes.
The long way across is the major axis and the short way is the minor axis. The farther apart the foci, the flatter the oval; when the foci merge into one point the oval becomes a circle. Planetary orbits, whispering galleries and some race tracks are ellipses.
Formula and notation
With center at the origin and the major axis along the x-axis:
\[b^2x^2+a^2y^2=a^2b^2,\quad a>b>0,\qquad c^2=a^2-b^2.\]
The vertices are \((\pm a,0)\), the foci are \((\pm c,0)\), and the eccentricity \(e=\frac{c}{a}\) lies between 0 and 1.
Key facts
- The constant distance sum equals 2a, the length of the major axis.
- The larger denominator in the equation sits under the variable along the major axis.
- Eccentricity near 0 looks almost circular; near 1 looks long and thin.
- Shifting the center to (h, k) replaces x and y by x − h and y − k.
Ellipse example
Find the vertices, foci and eccentricity of \(5x^2+9y^2=180\).
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Divide every term by 180 so the right side becomes 1. The denominators are then 36 under \(x^2\) and 20 under \(y^2\): \(a^2=36\), \(b^2=20\).
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The larger denominator sits under \(x\), so the major axis is horizontal with vertices at \(\pm 6\).
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Focal distance: \(c^2=36-20=16\), so \(c=4\) and the foci are at \(\pm 4\) on the x-axis.
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Eccentricity: four sixths, which is \(\frac{2}{3}\).
Vertices \((\pm 6,0)\), foci \((\pm 4,0)\), eccentricity \(\frac{2}{3}\).
Common mistake
Watch out
Lucia computes the focal distance for this oval as \(c=\sqrt{36+20}=\sqrt{56}\approx 7.5\).
Why it is wrong
For an ellipse the focal distance comes from a difference, \(c^2=a^2-b^2\). The sum belongs to the hyperbola. Her value 7.5 would place the foci outside the vertices at 6, which is impossible.
Related terms and lessons
Browse the glossary: A–Z list · Precalculus vocabulary · Geometry and Measurement