what is directrix.?? Write the equation for directrix of an ellipse????
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Answers
Equation
The equations of the directrices for ellipse are given below.
There are following two cases. (a and b are lengths of semi-major axis and semi minor axis)
Case I : When major axis is parallel to X-axis
In this case, the ellipse is said to be in its standard position. Under this category, there are two types of ellipses.
(A) Whose center is positioned at the origin
The equation of such ellipse is
x^2/a^2 + y^2/b^2 = 1
The equations of directrices are:
x = ae and x = - ae
B) Whose center is positioned at the point (h,k)
The equation of such ellipse would be
(x−h)^2/a^2 + (y−k)^2/b^2 = 1
The equations of directrices are:
x =h + a/e and x =h - a/e
Case II : When major axis is parallel to Y-axis
Similarly, there are two kinds of ellipses under this case.
(A) Whose center is positioned at the origin
The equation of such ellipse is given by
x^2/b^2 + y^2/a^2 = 1
The equations of directrices would be the same as above:
x = a/e and x = - a/e
B) Whose center is positioned at the point (h,k)
The equation of such ellipse is as follows
(x−h)^2/b^2 + (y−k)^2/a^2 z= 1
The equations of directrices are:
x = k + a/e and x = k - a/e
Steps
In order to find the equation of the directrices from the equation of an ellipse, the following steps should be adopted:
Step 1: Compare the given equation of ellipse with standard equation and find the values of a and b. Also, determine the values of h and k if center is not at origin.
Step 2: Use the formula of eccentricity for calculating the value of e. Its formula is given by:
e = a^2−b^2/a and e < 1.
Step 3: Now, substitute the above values in the formula (whichever is suitable) for the equations of the directrices of an ellipse.
Examples
Let us have a look at the following examples of directrices of ellipse.
Example 1: Find the equations of directrices of an ellipse whose equation is x^2/4 + y^2/1 = 1.
Solution: Given equation can be rewritten as
x^2/2^2 + y^2/1^2 = 1
Thus, a = 2 and b = 1 and it is centered at origin.
We have e = a^2−b^2/a
= 2^2−1^2/2
= 3/2
The equations of directrices are
x = a/e and x = - a/e
x = 2/(3/2 ) and x = - 2/(3/2)
x = 4/3 and x = - 4/3