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Directrix of Ellipse
Directrix of ellipse is a line parallel to the latus rectum of the ellipse and are perpendicular to the major axis of the ellipse. The ellipse has two directrices. The two directrix of ellipse are equidistanct from the center or the minor axis of the ellipse. The directrix is used to define the eccentricity of the ellipse.
Let us learn more about the directrix of the ellipse, its properties, and related terms, with the help of examples, FAQs.
What Is Directrix of Ellipse?
Directrix of ellipse is parallel to the latus rectum of the ellipse and is drawn outside the ellipse. The ellipse has two directrices which are perpendicular to the major axis of the ellipse. Directrix is used to define the eccentricity of ellipse: the ratio of distances of any point on the ellipse from the foci of ellipse and the directrix of an ellipse is the eccentricity of ellipse and it is lesser than 1. (e < 1). For the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), the equations of the two directrix of ellipse is x = +a/e, and x = a/e.
The directrix of the ellipse is equidistant from the center of the ellipse. The two directrices of the ellipse are parallel to each other and are also parallel to the minor axis of ellipse.
Properties of Directrix of Ellipse
The below points mentioning the properties of the ellipse help for a better understanding of the properties of the ellipse.
 The directrix of the ellipse is passing through the focus of the ellipse.
 The ellipse has two foci, and hence it has two directrices.
 The directrix of the ellipse is parallel to the latus rectum of the ellipse.
 The directrix of an ellipse is perpendicular to the major axis of the ellipse.
 The directrix of an ellipse is parallel and equidistant from the minor axis of the ellipse.
Terms Related to Directrix of Ellipse
The following terms are related to the directrix of ellipse and are helpful for easy understanding of the directrix of ellipse.
 Foci Of Ellipse: The ellipse has two foci that lie on the major axis of the ellipse. The coordinates of the two foci of the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) are (ae, 0), and (ae, 0). The foci of ellipse and the vertices of ellipse are collinear.
 Eccentricity of Ellipse: The ratio of the distance of a point on the ellipse from the foci of ellipse and the directrix of ellipse is called the eccentricity of ellipse and it is lesser than 1 (e < 1). The eccentricity of ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) is \(e = \sqrt {1  \dfrac{b^2}{a^2}}\).
 Major Axis of Ellipse: The line passing through the foci and the vertices of the ellipse is the major axis of the ellipse. For an ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) the length of the major axis of the ellipse is 2a units.
 Minor Axis of Ellipse: The line passing through the center of ellipse and the covertices of ellipse are the minor axis of ellipse. For an ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) the length of the minor axis of the ellipse is 2b units.
 Latus Rectum of Ellipse: The focal chord of the ellipse which is perpendicular to the axis of the ellipse is the latus rectum of ellipse. The ellipse has two foci and hence it has two latus rectums. The length of latus rectum of an ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) is 2b^{2}/a.
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Examples on Directrix of Ellipse

Example 1: Find the directrix of ellipse having the equation \(\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1\).
Solution:
The given equation of the ellipse is \(\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1\).
Comparing this with the standard equation of ellipse \(\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1\) we have \(a^2 = 25\), \(b^2 = 16\).
And we have a = 5, b = 4.
Let us first find the eccentricity of ellipse.
\(e = \sqrt {1  \dfrac{b^2}{a^2}}\)
\(e = \sqrt {1  \dfrac{16}{25}} \)
\(e = \sqrt {\dfrac{25  16}{25}}\)
\(e = \sqrt {\dfrac{9}{25}}\)
e = 3/5
The directrix of ellipse is x = +a/e.
x = +5/3/5
x = + 25/3
Therefore the diretrix of ellipse is x = +25/3, and x = 25/3 respectively.

Example 2: Find the equation of an ellipse given that the directrix of an ellipse is x = 8, and the focus is (2, 0).
Solution:
The given equation of directrix of ellipse is x = 8, and comparing this with the standard form of the equation of directrix x = + a/e, we have a/e = 8.
The given focus of ellipse is (ae, 0) = (2, 0), which gives us ae = 2.
Let us divide a/e with ae to find the value of e  the eccentricity of the ellipse.
\(\dfrac{a/e}{ae} = \dfrac{8}{2}\)
\(\dfrac{1}{e^2} = 4\)
1/e = 2
e = 1/2
ae = 2
a × 1/2 = 2
a = 4
Let us find the value of b, which is needed to find the equation of the ellipse.
\(b^2 = a^2(1  e^2)\)
\(b^2 = 4^2(1  (1/2)^2)\)
\(b^2 = 16(1  1/4)\)
\(b^2 = 16× 3/4\)
\(b^2 = 12\).
The required equation of ellipse is \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\)
\(\dfrac{x^2}{16} + \dfrac{y^2}{12} = 1\)
Therefore the required equation of the ellipse is \(\dfrac{x^2}{16} + \dfrac{y^2}{12} = 1\).
FAQs on Directrix of Ellipse
What Is the Directrix Of Ellipse In Geometry?
The directrix of ellipse is a line parallel to the latus rectum of ellipse and is perpendicular to the major axis of the ellipse. The ellipse has two directrices. The given equation of ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) has two directrix which are x = +a/e, and x = a/e.
How To Identify the Directrix Of Ellipse?
The directrix can be identified as the lines parallel to the latus rectum, and also the minor axis of the ellipse. The directrix of the ellipse are the lines drawn external to the ellipse and are perpendicular to the major axis of the ellipse.
How To Find Directrix Of Ellipse From The Equation Of Ellipse?
The directrix of the ellipse can be derived from the equation of the ellipse in two simple steps. The equation of the ellipse is \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\). From this we can find the value of 'a' and also the eccentricity 'e' of the ellipse. Thus the required equation of directrix of ellipse is x = +a/e, and x = a/e.
How Many Directrix Does An Ellipse Have?
The ellipse has two directrices, similar to the two foci of the ellipse.
What Are The Applications Of the Directrix Of Ellipse?
The directrix of ellipse can be used to find the coordinates of the focus, the eccentricity of ellipse, and also the equation of the ellipse.
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