Latus Rectum
Latus rectum is a line passing through the foci of the conic and is parallel to the directrix of the conic. The latus rectum is the focal chord and the number of latus rectums is equal to the number of foci in the conic. A parabola has one latus rectum, and an ellipse, hyperbola has two latus rectums.
Let us learn more about each of the latus rectums of parabola, ellipse, hyperbola, their lengths, and the endpoints of the latus rectums.
What Is a Latus Rectum?
The latus rectum is a line passing through the foci of the conic and is parallel to the directrix of the conic. The latus rectum is also the focal chord which is perpendicular to the axis of the conic. The parabola has one latus rectum, but the ellipse and the hyperbola have two latus rectum since it has two foci. Here we shall learn more about the following three latus rectums.
- Latus Rectum of a Parabola
- Latus Rectum of an Ellipse
- Latus Rectum of a Hyperbola
Latus Rectum of a Parabola
The latus rectum of a parabola is the chord that is passing through the focus of the parabola and is perpendicular to the axis of the parabola. The latus rectum of parabola can also be understood as the focal chord which is parallel to the directrix of parabola. The length of latus rectum for a standard equation of a parabola y2 = 4ax is equal to LL' = 4a. The endpoints of the latus rectum of a Parabola are (a, 2a), (a, -2a).
The endpoints of the latus rectum of a parabola and the focus of the parabola are all collinear. The distance between the endpoints of the latus rectum is equal to the length of latus rectum.
Latus Rectum of a Ellipse
The latus rectum of an ellipse is a line passing through the foci of the ellipse and is drawn perpendicular to the transverse axis of the ellipse. The latus rectum of ellipse is also the focal chord which is parallel to the directrix of the ellipse. The ellipse has two foci and hence the ellipse has two latus rectums. The length of the latus rectum of the ellipse having the standard equation of x2/a2 + y2/b2= 1, is 2b2/a.
The endpoints of the latus rectum of the ellipse passing through the focus (ae, 0), is (ae, b2/a), and (ae, -b2/a). And the endpoints of the latus rectum of the ellipse passing through the foci (-ae, 0), is (-ae, b2/a), and (-ae, -b2/a). Here 'e' is the eccentricity of the ellipse and its value lies between 0 and 1, (0 < e < 1). The endpoints of the latus rectum of the ellipse and the focus of the ellipse are collinear, and the distance between the endpoints of the latus rectum gives the length of latus rectum.
Latus Rectum of a Hyperbola
The latus rectum of a hyperbola is a line passing through the foci of the hyperbola and is drawn perpendicular to the transverse axis of the hyperbola. The latus rectum of a hyperbola is also the focal chord which is parallel to the directrix of the ellipse. The hyperbola has two foci and hence the hyperbola has two latus rectums. The length of the latus rectum of the hyperbola having the standard equation of x2/a2 - y2/b2= 1, is 2b2/a.
The endpoints of the latus rectum of the hyperbola passing through the focus (ae, 0), is (ae, b2/a), and (ae, -b2/a). And the endpoints of the latus rectum of the hyperbola passing through the focus (-ae, 0), is (-ae, b2/a), and (-ae, -b2/a). Here 'e' is the eccentricity of the hyperbola and is always greater than 1, (e > 1). The endpoints of the latus rectum of the hyperbola and the focus of the hyperbola are collinear, and the distance between the endpoints of the latus rectum gives the length of latus rectum.
Related Topics
The following topics will help in a better understanding of the latus rectum.
FAQs on Latus Rectum
What Is a Latus Rectum?
A latus rectum is a straight line passing through the focus of the parabola and is perpendicular to the axis of the parabola. The latus rectum of the parabola is the focal chord which is parallel to the directrix of a parabola. The prabola has only one latus rectum, but the ellipse and hyperbole have two latus rectums.
How Do You Find the Length of the Latus Rectum?
The length of the latus rectum is equal to the distance between the two endpoints of the latus rectum. The length of the latus rectum of a parabola is 4a, and the length of the latus rectum of an ellipse, and a hyperbola is equal to 2b2/a.
What Is the Length Of Latus Rectum?
The length of latus rectum of a parabola is 4a, and the length of the latus rectum of an ellipse or a hyperbola is equal to 2b2/a. These lengths are with reference to the standard form of equations of the parabola, ellipse, or hyperbola.
What Are the End Points of a Latus Rectum?
The endpoints of the latus rectum of a Parabola y2= 4ax are (a, 2a), (a, -2a). The endpoints of the latus rectum of the ellipse passing through the focus (ae, 0), is (ae, b2/a), and (ae, -b2/a). And the endpoints of the latus rectum of the ellipse passing through the foci (-ae, 0), is (-ae, b2/a), and (-ae, -b2/a). And the endpoints of the latus rectum of the hyperbola are the same as the endpoints of the latus rectum of an ellipse.
Does a Hyperbola Have a Latus Rectum?
The hyperbola also has a latus rectum. The hyperbola has two foci and hence it has two latus rectums.
What Are The Uses Of Latus Rectum?
The latus rectum is useful to find the directrix of the conic, which in turn helps to find the eccentricity and the equation of the respective conic.