Directrix of Ellipse

An ellipse is defined as the locus of all points in a plane such that the sum of their distances from two fixed points, is constant. Base on the definition, the elliptical equation could be derived in various forms to represent the trajectory of the curve. For example, the standard form in rectangular coordinates.

\dfrac{x^2}{a^2} +\dfrac{y^2}{b^2}=1

(1)

where a is the semi-major axis and b is the semi-minor axis.

An ellipse could also be defined as the locus of points such that the ratio of the distance of any point on the curve from a fixed point and from a fixed line is constant. The fixed line is called directrix. It is a line that is perpendicular to the major axis and parallel to the latus rectum of the ellipse. The question - what is the equation of directrix for an ellipse? How to find the equation of directrix mathematically?

In the figure is an ellipse with its semi-major axis a, semi-minor axis b and the distance from foci to the center c. Since the directrix is parallel to the minor axis, let d be the distance from the origin to the directrix line. Then the equation of directrix is,

x =d

(2)

Now let's establish equations to solve for d.

According to the definition, the ratio of the distance of any point P from focus F2 and the directrix is constant. We are going to use two special points. One is the intersection of the positive x axis with the ellipse A(a,0). Another is the intersection of positive y axis with the ellipse B(0,b). Let e be the ratio of the distance from focus and directrix. Then, for point A,

e = \dfrac{a-c}{d-a}

(3)

For point B,

e =\dfrac{BF2}{d} = \dfrac{\sqrt{c^2+b^2} }{d}

(4)

For an ellipse, there's the following equation

a^2 = c^2+b^2

(5)

Substitute (5) into (4), the equation (4) is simplified as

e = \dfrac{\sqrt{a^2} }{d} = \dfrac{a}{d}

(6)

An equation is established using (3) and (6).

\dfrac{a-c}{d-a} = \dfrac{a}{d}

Cross-multiplying gives,

(a-c)d = a(d-a)

(a-c)d = ad-a^2

cd = a^2

d = \dfrac{a^2}{c}

(7)

Since an ellipse has two directices, the equation of the other directrix could be derived using the symmetrical feature of the ellipse. The combined equation is as follow,

d =\pm \dfrac{a^2}{c}

(8)

Plug (7) into (6), we get the formula for e.

e = \dfrac{a}{d}=a\cdot \dfrac{c}{a^2} = \dfrac{c}{a}

(9)

which is the eccentricity. Since c< a, the eccentricity of an ellipse is always less than 1. And the distance from the directrix to origin is larger than a .

d = \dfrac{a^2}{c} >a

which means a directrix is always outside the ellipse.

On the other hand, substituting (9) into (7) gives

d = \dfrac{a}{e}

which shows that the distance from the directrix of an ellipse to its origin is inversely proportional to the eccentricity of the ellipse. The smaller the eccentricity, the longer the distance from the directrix to origin

Summary about the directrix of ellipse

The directrix of ellipse is a line that is parallel to the latus rectum of the ellipse and used to determine the eccentricity of the ellipse.
The ratio of the distance of any point on an ellipse from its focus and the directrix is constant.
There are two directrices that are symmetrical to minor axis of the ellipse and always outside the ellipse.
The distance from origin to directrix isinversely proportional to the eccentricity of the ellipse.

Collected in the board: Conic sections

Steven Zheng posted 1 year ago