The focus of an ellipse is at the origin. The directrix is the line x = 4 and its eccentricity is $12$ then length of its semi major axis is:

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answer is D.

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Detailed Solution

Let us assume that the focus as S(0,0).
We are given that the equation of directrix as x = 4
We know that if the directrix is parallel to the Y-axis then the coordinate axes are the axes of the ellipse.
Let us take the rough figure of the given ellipse as follows:

https://www.vedantu.com/question-sets/05bd8688-374f-4e01-b6fd-ee986e0e2ec96074019770203862776.png

We know that the focus for the standard equation of ellipse

x 2 a 2 + y 2 b 2 = 1

is at

(a e, 0)

.
Now, let us use the transformation of axes that is let us transform the given ellipse such that the focus becomes

(a e, 0)

Now, let us take the equation of ellipse according to transformation of axes as

X 2 a 2 + Y 2 b 2 = 1

Here, we can see that we increased the X coordinate by

a e

to get the new equation
By converting the above equation to mathematical equation then we get
⇒x + ae = X
⇒x = X − ae
We are given that the equation of directrix as x = 4 for old ellipse
By converting the above equation to new form of ellipse then we get
⇒ X − ae = 4
⇒ X – ae = 4
Now, let us find the distance between focus and directrix of the new equation
Let us assume that the distance between centre and directrix as d
We know that the formula of distance of point (h, k) to the line