The latus rectum of an ellipse

An ellipse is defined as the set of points in a plane such that the sum of the distances from two fixed points is constant. The two fixed points are called foci (focus for each) of the ellipse. The latus rectum of the ellipse is a chord that passes a focus and is perpendicular to the major axis. Since an ellipse is symmetrical about both its major and minor axis, there are two such chords that are parallel to each other and the same in length in the ellipse.

Length of the latus rectum of an ellipse

In Cartesian coordinates, an ellipse centered at origin and with its major axis and minor axis on the axis x and y, could be expressed as,

\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. Let c be the half of the distance between two foci of the ellipse. Then we have the following identity for an ellipse.

a^2 = b^2+c^2

(2)

We are going to show that the length of a latus rectum could be represented in terms of the length of semi-axises.

First, plot a line segment that passes F2 and is parallel to axis y. Then the line segment has two has two intersecting points that are symmetric about the major axis. Let the y coordinate of one point is l. Then the coordinates of both points are determined as (c,l) and (c,-l). The length of the chord L will be,

L= 2l

(3)

Secord, plug one of points into the ellipse equation (1) and solve for l

\dfrac{c^2}{a^2} +\dfrac{l^2}{b^2}=1

Move the term of constent to right hand side

\dfrac{l^2}{b^2}=1 - \dfrac{c^2}{a^2} = \dfrac{a^2-c^2}{a^2}

(4)

Use the identity (2) for (4), and then

\dfrac{l^2}{b^2} =\dfrac{b^2}{a^2}

Taking square root yields

l = \dfrac{b^2}{a}

(5)

And subsequently,

L =\dfrac{2b^2}{a}

(6)

Now we have derived the formula for the length of the latus rectum of an ellipse. Since b < a,

L = \dfrac{2b\cdot b }{a} < \dfrac{2b\cdot a }{a} = 2b

which shows the length of the latus rectum of an ellipse is always less than that of minor axis.

Latus rectum in Polar Coordinates

An ellipse could also be represented in polar coordinates by an equation in terms of radius and angle that is formed by the radius with major axis. If the polar coordinate is centered at F1, the equation of the ellipse is given as

r = \quad = \dfrac{l}{1-e \cos\theta }

(7)

where l =\dfrac{b^2}{a} , which is the semi-latus rectum of the ellipse and e = \dfrac{c}{a} , which is the eccentricity of the ellipse.

Summary about the latus rectum of an ellipse

The latus rectum of the ellipse is a chord that passes a focus and is perpendicular to the major axis.
There are two latus rectums that are symmetric about minor axis.
The length of the latus rectum of an ellipse is \dfrac{2b^2}{a}, which is always less than that of minor axis.
The equation of an ellipse in Polar Coordinates could be expressed in terms of the latus rectum and the eccentricity of the ellipse.

Collected in the board: Conic sections

Steven Zheng posted 1 year ago