Respuesta :
The each directrix are 33.85 units from the center on the major axis option (C) horizontal line that is 33.8 units is correct.
It is given that an ellipse is represented by the equation given below:
[tex]\rm \frac{(y-3)^2}{169} + \frac{(x+6)^2}{144} = 1[/tex]
It is required to find the correct option given in the question:
What is an ellipse?
It is defined as the curve in which there is two points on the plane and these points distannce from fixed(focal point) is a constant value.
We have an ellipse equation:
[tex]\rm \frac{(x+6)^2}{144}+\frac{(y-3)^2}{169} = 1[/tex]
We know the standard equation of an ellipse is given by:
[tex]\rm \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1[/tex]
On comparing the given equatiion with standard equation, we get:
[tex]\rm a^2 = 144\\[/tex] ⇒ a = 12
[tex]\rm b^2 = 169[/tex] ⇒ b = 13
and the center of the ellipse is (h,k) : (-6,3)
We know the distance from centre to focus is given by:
[tex]\rm c^2= b^2-a^2[/tex]
[tex]\rm c^2 = 169-144 \Rightarrow25[/tex]
c = 5
and the eccentricity (e) of an ellipse:
[tex]\rm e=\frac{c}{b}[/tex]
[tex]\rm e=\frac{5}{13}[/tex] ⇒ 0.384
Directrix of an ellipse:
[tex]\rm y=\pm \frac{b}{e}[/tex]
[tex]\rm y=\pm \frac{13}{0.384}\\\\\rm y = \pm \ 33.85[/tex]
Thus, the each directrix are 33.85 units from the center on the major axis option (C) horizontal line that is 33.8 units is correct.
Learn more about the ellipse here:
https://brainly.com/question/19507943
