Math101 · Independent Ontario learning library
Math101Ellipse
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 $2a$.
