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AlladinOne [14]
10 months ago
8

In a solar system, two comets pass near the sun, which is located at the origin. Comet E is modeled by quantity y plus 16 end qu

antity squared over 400 plus x squared over 144 equals 1 and Comet H is modeled by quantity y plus 13 end quantity squared over 144 minus x squared over 25 equals 1 comma where all measurements are in astronomical units.Part A: What are the vertices for the path of Comet E? Show your work. (4 points)Part B: Which comet travels closer to the sun? Give evidence to support your answer. (5 points)Part C: What key feature does the sun represent for both comets? Give evidence to support your answer. (6 points)Link to view the question file:///C:/Users/funkm/Downloads/Untitled%20document%20(2).pdf
Mathematics
1 answer:
Leya [2.2K]10 months ago
5 0

Explanation:

The equation of comet E is given below as

\frac{(y+16)^2}{400}+\frac{x^2}{144}=1

The equation of comet H is modelled below as

\frac{(y+13)^2}{144}-\frac{x^2}{25}=1

Given, that the Sun is located at the origin.

This is an equation of an ellipse. So, the path traveled by Comet e is an elliptic path.

Comparing with the Standard equation of ellipse: below,we will have

\frac{(y-k)^2}{b^2}+\frac{(x-h)^2}{a^2}

Whise center is

\begin{gathered} center=(h,k) \\ vertices=(h\pm a,0) \end{gathered}

By comparing coefficient, we will have

\begin{gathered} a^2=144 \\ a=12 \\ b^2=400 \\ b=20 \\ k=-16 \\ h=0 \\ (h,k)=(0,-16) \end{gathered}

Hence,

The vertex of comet E will be

\begin{gathered} (h\pm a,0) \\ (0+12,0),(0-12,0) \\ (12,0),(-12,0) \end{gathered}

The vertex of comet E is

(12,0),(-12,0)

Part C:

It is the foci forboth comets

\begin{gathered} cometE: \\ c=\sqrt{b^2-a^2} \\ c=\sqrt{400-144} \\ c=\sqrt{256} \\ c=16 \\  \\ For\text{ comet F:} \\ c=\sqrt{a^2-b^2} \\ c=\sqrt{144+25} \\ c=\sqrt{169} \\ c=13 \end{gathered}

Hence,

The foci will be

\begin{gathered} (16\pm16,0) \\ cometE \\ (0,0),(32,0) \\ cometF: \\ (13\pm13,0) \\ (0,0),(26,0) \end{gathered}

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