Concepts: the ellipse
Conic Sections24 min readLesson 4 of 7
4.1 Definition and standard equation
An ellipse is the set of points whose distances from two fixed points, the foci, add up to a constant. The standard equation x squared by a squared plus y squared by b squared equal to 1, with a greater than b, has foci at (plus or minus c, 0), where c squared equals a squared minus b squared. The constant sum of distances is 2a, the length of the major axis is 2a and the length of the minor axis is 2b.
The centre is the origin and the axes lie along the coordinate axes. If b is greater than a, the major axis lies along the y-axis instead and the foci are (0, plus or minus c) with c squared equal to b squared minus a squared. Read which semi-axis is larger before writing any focus.
The ellipse fits inside the rectangle with sides 2a and 2b. This rectangle is a quick sketch aid: the foci always lie inside the major axis, the distance c is smaller than a, and the eccentricity c by a is between 0 and 1.
Directrix questions are quick once the eccentricity is known. For an ellipse the directrices are at x equal to plus or minus a by e, so the distance between them is 2a by e; knowing any two of a, e and that distance fixes the third. The same quantity appears in the standard derivation of the ellipse from focus and directrix, which is why exercise lists directrix questions alongside eccentricity questions, and why the two should be revised together.
4.2 Eccentricity, directrices and latus rectum
The eccentricity of an ellipse is c by a and measures how far the shape is from a circle. For a circle the two foci coincide, c is zero and the eccentricity is zero; as the ellipse gets longer, the eccentricity increases toward 1. The directrices are the lines x equal to plus or minus a by e, and the distance between them is 2a by e.
A point on the ellipse can be written as (a cos theta, b sin theta). This parametric form is useful when a question gives a point whose coordinates involve cosines and sines, because substituting simplifies immediately. The latus rectum is 2b squared by a, the chord through a focus perpendicular to the major axis.
For a shifted ellipse centred at (h, k), the equation is (x minus h) squared by a squared plus (y minus k) squared by b squared equal to 1. The major and minor axes keep their lengths, only the centre moves, so the foci and directrices shift with the centre.
When a question gives a point on the ellipse with coordinates as cosines and sines, substitute directly into the standard equation before doing anything else. The trigonometric identity then removes the parameter, and the remaining condition is usually a single equation in the semi-axes.
4.3 Exercise 10.3 practice
The first item of the ellipse exercise uses x squared by 36 plus y squared by 16 equal to 1 and asks for the foci, the vertices, both axis lengths and the eccentricity. Here a is 6 and b is 4, so c squared is 36 minus 16 equal to 20 and c is 2 root 5. The foci are (plus or minus 2 root 5, 0), the vertices are (plus or minus 6, 0), the major axis is 12, the minor axis is 8 and the eccentricity is root 5 by 3.
Question 3 asks for the ellipse with vertices at (plus or minus 5, 0) and foci at (plus or minus 4, 0). Then a is 5 and c is 4, so b squared is 25 minus 16 equal to 9 and the equation is x squared by 25 plus y squared by 9 equal to 1.
Question 8 asks for the eccentricity of x squared by 25 plus y squared by 9 equal to 1. Here a is 5 and b is 3, c squared is 16, c is 4 and the eccentricity is 4 by 5.
Directrix questions are quick once the eccentricity is known. For an ellipse the directrices are at x equal to plus or minus a by e, so the distance between them is 2a by e; knowing any two of a, e and that distance fixes the third. The same quantity appears in the standard derivation of the ellipse from focus and directrix, which is why the exercise lists directrix questions alongside the eccentricity questions.
Tip
Compare a and b before writing the foci. If b is larger, the major axis is vertical and the focus coordinate is on the y-axis.