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Latus Rectum Of Ellipse

Latus rectum of ellipse is a straight line passing through the foci of ellipse and perpendicular to the major axis of ellipse. Latus rectum is the focal chord, which is parallel to the directrix of the ellipse. The ellipse has two foci and hence it has two latus rectums. The length of latus rectum of ellipse x2/a2 + y2/b2= 1, is 2b2/a.

Each of the latus rectum cuts the ellipse at two distinct points. Let us learn more about the properties of latus rectum of ellipse, with the help of examples, and FAQs.

1. What Is Latus Rectum Of Ellipse?
2. Properties Of Latus Rectum Of Ellipse
3. Terms Related To Latus Rectum Of Ellipse
4. Examples On Latus Rectum Of Ellipse
5. Practice Questions
6. FAQs On Latus Rectum Of Ellipse

What Is Latus Rectum Of 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 an 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.

Latus Rectum Of Ellipse

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 the latus rectum.

Properties Of Latus Rectum of Ellipse

The important properties of the latus rectum of the ellipse are as follows.

  • The latus rectum is perpendicular to the major axis of the ellipse.
  • The latus rectum of the ellipse passes through the focus of the ellipse.
  • There are two latus rectums for an ellipse.
  • Each latus rectum cuts the ellipse at two distinct points.
  • The latus rectum is parallel to the directrix of the ellipse.

The following terms are related to the latus rectum of the ellipse and help for a better understanding of the concept of the latus rectum of the ellipse.

  • Foci of Ellipse: The focus of the ellipse lies on the major axis of the ellipse. The ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1\) has two foci and their coordinates is (+ae, 0), and (-ae, 0). The midpoint of the foci of the ellipse is the center of the ellipse.
  • Focal Chord: The line passing through the focus of the ellipse is the focal chord of the ellipse. The ellipse has an infinite number of focal chords passing through the focus. The focal chord perpendicular to the axis of the ellipse is called the latus rectum of the ellipse.
  • Directrix: Directrix is a line that is drawn outside the ellipse and is perpendicular to the major axis of the ellipse. Directrix is useful to give the definition of an ellipse. The ellipse is the locus of a point such that the ratio of its distances from the focus and the directrix is less than 1.
  • Vertex of Ellipse: A vertex of an ellipse is the point of intersection of the ellipse with its axis of symmetry. The ellipse intersects its axis of symmetry at two distinct points, and hence an ellipse has two vertices. The vertex of an ellipse is also the point of intersection of the line which is passing through the foci of the ellipse and is cutting the ellipse at two distinct points.
  • Major Axis of Ellipse: The major axis of the ellipse is a line that cuts the ellipse into two equal halves. The major axis is a line passing through the foci and the center of the ellipse. For an ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1\), the endpoints of the major axis are (a, 0), (-a, 0), and the length of the major axis is 2a units.
  • Minor Axis of Ellipse: The minor axis of the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1\), is the axis that is perpendicular to its major axis. The endpoints of the minor axis of the ellipse is (0, b), (0, -b), and the length of the minor axis is 2b units. The minor axis also passes through the center of the ellipse.

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Examples On Latus Rectum Of Ellipse

  1. Example 1: Find the length of the latus rectum of the ellipse \(\dfrac{x^2}{36} + \dfrac{y^2}{25}\) = 1.

    Solution:

    The given equation of the ellipse is \(\dfrac{x^2}{36} + \dfrac{y^2}{25}\) = 1.

    Comparing this with the standard equation of the ellipse \(\dfrac{x^2}{36} + \dfrac{y^2}{25}\) = 1 we have a^2 = 36, and b^2 = 25.

    And we have a = 6, and b = 5.

    The formula for the length of the latus rectum is 2b2/a.

    Length of latus rectum = 2×52/6 = 25/3

    Therefore, the length of latus rectum of ellipse is 25/3 units.

  2. Example 2: Find the end points of the latus rectum of the ellipse \(\dfrac{x^2}{64} + \dfrac{y^2}{49}\) = 1.

    Solution:

    The given equation of the ellipse is \(\dfrac{x^2}{25} + \dfrac{y^2}{16}\) = 1

    This can be compared with the standard equation of the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}\) = 1, and we have \(a^2\) = 25, \(b^2\) = 16.

    We get a = 5, and b = 4.

    Let us first calculate the eccentricity of the ellipse using the below formula.

    \(e = \sqrt {1 - \dfrac{b^2}{a^2}}\)

    \(e = \sqrt {1 - \dfrac{^42}{5^2}}\)

    \(e = \sqrt {\dfrac{25 - 16}{25}}\)

    \(e = \sqrt {\dfrac{9}{25}}\)

    \(e = 3/5\)

    The endpoints of the latus rectum of the ellipse are (ae, b2/a), (-ae, b2/a), (-ae, -b2/a), and (ae, -b2/a). Substituting the values of a and b we can obtain the endpoints of the latus rectum.

    (ae, b2/a) = (5× 3/5, 42/5) = (3, 16/5)

    (-ae, b2/a) = (-5× 3/5, 42/5) = (-3, 16/5)

    (-ae, -b2/a) = (-5× 3/5, -42/5) = (-3, -16/5)

    (ae, -b2/a) = (5× 3/5, -42/5) = (3, -16/5)

    Therefore the end points of the latus rectums are (3, 16/5), (-3, 16/5), (-3, -16/5), and (3, -16/5).

    Thus the end points

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FAQs on Latus Rectum Of Ellipse

What Is Latus Rectum Of Ellipse In Geometry?

Latus rectum of ellipse is the focal chord that is perpendicular to the axis of the ellipse. The ellipse has two focus and hence there are two latus rectum for an ellipse. The length of the latus rectum of the ellipse having the standard equation of x2/a2 + y2/b2= 1, is 2b2/a.

What Is The Length Of the Latus Rectum Of Ellipse?

The length of the latus rectum of ellipse is the distance between the endpoints (ae, b2/a), and (ae, -b2/a), and is equal to 2b2/a.

What Are End Points Of Latus Rectum Of Ellipse?

There are two latuc rectums of the ellipse. The endpoints of the latus rectum passing through the focus (+ae, 0) is (ae, b2/a) and (ae, -b2/a). And the endpoints of the latus rectum passing through the focus (-ae, 0) is (-ae, b2/a), and (-ae, -b2/a).

How Many Latus Rectums Does An Ellipse Have?

The ellipse has to foci, and has two latus rectus passing through these foci.

What Is The Relationship Between Latus Rectum Of Ellipse And Foci Of Ellipse?

The latus rectum of ellipse is a chord that passes through the foci of an ellipse. The ellipse has two foci and hence there are two latus rectum of an ellipse. And the latus rectum is the focal chord which is perpendicular to the major axis passing through the foci of the ellipse.

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