Overview on Latus Rectum of Parabola

The conical sections have various components such as the focus, directrix, latus rectum, etc. All these components together define the conic section. The latest rectum of the conic search section is a call that passes through the focus parallel to the conic section directrix.

The word latus rectum is derived from the Latin word latus and rectum, which means side straight.

The practice rectum is perpendicular to the major axis. Both of the endpoints of the latus rectum are on the curve. The latus rectum is a line segment of a specific type on a conical section.

Latus Rectum

As discussed earlier, the letter’s rectum is a line segment that passes through the foci of the conical sections. The rectum is also parallel to the directrix of the conical section. The number of lattice rectums present in a conical section is equal to the number of foci. For example, an ellipse and hyperbola both have two latus rectums, but a parabola has one latus rectum.

The latus rectum formula for each of these conical sections is different. The length of the latus rectum is also before for each conical section. 

Following are the lengths of the latus rectum for each conical section.

The latus rectum of the parabola is of the length which is four times its focal length.

The latus rectum of a hyperbola is twice the square length of the transverse axis which is divided by the conjugate axis.

The length of any latus rectum of an ellipse is twice the square length of the minor axis which is further divided by the length of the major axis.

The length of the latus rectum in a circle is always equal to its diameter’s length.

Let us learn about the latus rectums in various conical sections.

The Latus Rectum in a Parabola

In a parabola, the latus rectum is a line segment that passes through the focus of the parabola and is perpendicular to the axis of the parabola. The latus rectum in a parabola can also be regarded as the focal called that parallel to the directrix of the parabola. The latus rectum formula for a parabola with the equation y² = 4ax is equal to 4a. The endpoints of the latus rectum in a parabola will be (a, 2a) and  (a, -2a). 

In a parabola, the focus of the parabola and the endpoints of the latus rectum are collinear. The length of the latus rectum is the distance between the two endpoints of the latus rectum.

The Latus Rectum of the Hyperbola

In a hyperbola, the latus rectum is a line segment that passes through both the foci of the hyperbola and is perpendicular to the transverse axis of the hyperbola. The focal chord that runs parallel to the directrix of the ellipse can also be called the latus rectum of the hyperbola. The number of latus rectums in a conical section depends upon the number of foci. Since the hyperbola has two foci, there will be two latus recta in the hyperbola. 

The latus rectum formula for hyperbola has the standard equation,

x² / a² = y² / b² = 1 will be equal to 2b² /a.

The points (ae, b² /a ) and (ae, -b² /a ) are the endpoints that pass through the focus (ae, 0) of the hyperbola. Like in parabola, in hyperbola also, the endpoints of the latus rectum and the focus of the hyperbola are collinear. 

The Latus Rectum of the Ellipse 

In an ellipse, the latus rectum is the chord that passes through both the foci of the ellipse and is perpendicular to the transverse axis of the ellipse. The latus rectum of the ellipse also runs parallel to the directrix of the ellipse. The latus rectum formula for the ellipse with the standard equation x² / a² + y² / b² = 1 will be 2b² / a.

The (ae, b²/a) and (ae, -b²/a) are the endpoints of the latus rectum in an ellipse that passes through the focus (ae, 0).

The focus of the ellipse is collinear with the latus rectum of the ellipse. Since the number of latus rectum in a conical section depends upon the focus points of the conical section, the number of latus rectum in an ellipse is equal to 2.

Conclusion

The conic sections can be defined with the help of various components such as its foci, its directrix, its latus rectum, etc. The latus rectum of any conical section is the line segment that passes through the foci of the conic section and is parallel to the directrix of the conic section.

The latus rectum formula for each conic section, such as the parabola, the hyperbola, and the ellipse, are different. The number of latus rectums present for a particular conic section depends upon the number of focus points it has. The number of the latus rectum is equal to the number of foci.