Answer:
A) (-4, 0) and (36, 0)
B) Comet H
C) Focus
Step-by-step explanation:
<h3><u>Part A</u></h3>
<u>Given equation for Comet E</u>
The path of Comet E has been modeled as a horizontal ellipse.
<u>General equation of a horizontal ellipse</u>
where:
- center = (h, k)
- Vertices = (h±a, k)
- Co-vertices = (h, k±b)
- Foci = (h±c, k) where c²=a²-b²
- 2a = Major Axis: longest diameter of an ellipse
- 2b = Minor Axis: shortest diameter of an ellipse
- a = Major radius: one half of the major axis
- b = Minor radius: one half of the minor axis
Comparing the general equation with the given equation:
Therefore, the <u>vertices</u> are:
<h3><u>
Part B</u></h3>
<u>Given equation for Comet H</u>:
The path of Comet H has been modeled as a horizontal hyperbola.
<u>General equation of a horizontal hyperbola</u> (opening left and right):
where:
- center = (h, k)
- Vertices = (h±a, k)
- Co-vertices = (h, k±b)
- Foci = (h±c, k) where c²=a²+b²
- Transverse axis: y = k
- Conjugate axis: x = h
Comparing the general equation with the given equation:
Therefore, the <u>vertices</u> are:
As the sun is located at the origin (0, 0) the comet that travels closer to the sun is the comet whose vertex is closest to (0, 0).
Therefore, Comet H travels closer to the sun, since one of its vertex (-1, 0) is the closest to (0, 0).
<h3><u>Part C</u></h3>
The sun represents one of the foci of both Comet E and Comet H.
<u>Foci of Comet E</u>
<u>Foci of Comet H</u>
Learn more about ellipses here:
brainly.com/question/28152904
Learn more about hyperbolas here:
brainly.com/question/28164074