Math101 · Independent Ontario learning library
Math101learn.math101.caEllipse
An ellipse is the set of points whose sum of distances to two fixed foci is constant. In standard axis-aligned form $(x-h)^2/a^2+(y-k)^2/b^2=1$, the larger denominator identifies the major semi-axis.
Ellipses model planetary orbits, acoustics, optics, and architectural curves. The focal definition explains their reflection property.
Intuition and core definition
An ellipse is the set of points whose sum of distances to two fixed foci is constant. In standard axis-aligned form $(x-h)^2/a^2+(y-k)^2/b^2=1$, the larger denominator identifies the major semi-axis. A circle is the special case with equal semi-axes and coincident foci.
Notation, language, and conditions
The centre is $(h,k)$. Convention often uses $a\ge b>0$ regardless of orientation and $c^2=a^2-b^2$, with foci $c$ units from the centre along the major axis. Vertices are $a$ units along the major axis; co-vertices are $b$ units along the minor axis.
Why this idea matters
An ellipse combines a constant sum-of-distances property with unequal perpendicular semi-axes and two interior foci.
A dependable method
- Convert the equation to standard form equal to $1$.
- Read centre from shifted squares and identify the larger denominator.
- Take square roots for semi-axis lengths.
- Compute $c=\sqrt{a^2-b^2}$ and place foci along the major axis.
- Plot vertices and co-vertices and check the focal-distance sum $2a$.
Worked example
Representations and interpretation
The string-and-pins construction keeps the sum of focal distances fixed while a pencil traces the curve. Standard form shows the enclosing rectangle with half-width and half-height equal to the semi-axes.
Reasoning about variations
The larger denominator determines orientation, not the letter beneath it. If denominators are equal, $c=0$ and the ellipse becomes a circle. Translating the centre changes coordinates but not shape.
Common mistakes
How to check your work
- Substitute each vertex into the equation.
- Confirm $0\le c<a$ and foci lie inside the ellipse.
- Test the focal-distance sum at a vertex.
Practice
- For $x^2/36+y^2/20=1$, find $c$.
- State the centre of $(x+3)^2/4+(y-2)^2/16=1$.
- Which direction is its major axis?
Answers and brief solutions
Show answers
- $4$ $c=\sqrt{36-20}=4$.
- $(-3,2)$ Signs reverse inside squared groups.
- Vertical The larger denominator $16$ is under the y-term.
Synthesis and transfer
In an elliptical whispering gallery, rays leaving one focus reflect toward the other; the focal relationship links the physical effect to the standard equation parameters.
For $x^2/25+y^2/9=1$, the semi-major axis is $5$, the semi-minor axis is $3$, and $c=\sqrt{25-9}=4$. The foci are $(\pm4,0)$, and the sum of distances from any point on the curve to them is $10$. At the vertex $(5,0)$, those distances are $1$ and $9$, providing a direct check. If the two denominators become equal, $c$ becomes zero and the ellipse specializes to a circle. Increasing eccentricity moves the foci outward relative to the major axis and makes the curve less circular without creating open branches.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
For $x^2/36+y^2/20=1$, find $c$.
- $c=\sqrt{36-20}=4$.
End of lesson
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