Question

In a solar system, two comets pass near the sun, which is located at the origin. Comet E is modeled by quantity (y+16)^2/400 + x^2/144 = 1 and Comet H is modeled by quantity (y + 13)^2/144 - x^2/25 = 1, 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)

Answer 1

ANSWER and EXPLANATION

(a) For the path of the two comets, the sun is located at the origin (0, 0).

The equation that models the path of comet E is:

`((x+16)^2)/(400)+(y^2)/(144)=1`

This is the equation of an ellipse, in other words, the path of comet E is elliptic.

The standard equation of an ellipse is:

`((x-h)^2)/(b^2)+((y-k)^2)/(a^2)=1`

where (h, k) = center

(h, k ± a) = vertex

Comparing the equation for comet E to the general equation, we see that center of the ellipse is (0, -16) and the vertices of comet E are:

`\begin{gathered} (0,-16\pm20) \\ \\ (0,-16-20)\text{ and }(0,-16+20) \\ \\ (0,-36)\text{ and }(0,4) \end{gathered}`

The equation that models the path of comet H is:

`((y+13)^2)/(144)-(x^2)/(25)=1`

This is the equation of a hyperbola, hence, the path of comet H is hyperbolic.

The standard equation of a hyperbola is:

`((x-h)^2)/(a^2)-((y-k)^2)/(b^2)=1`

where (h, k) = center

(h, k ± a) = vertex

Comparing the equation of comet H to the standard equation, we see that the center of comet E is (0, 13), and the vertices of comet H are:

`\begin{gathered} (0,-13\pm12) \\ \\ (0,-13-12)\text{ and \lparen}0,-13+12) \\ \\ (0,-25)\text{ and }(0,-1) \end{gathered}`

(b) Checking the vertices of both comets, we see that the vertices of comet H are closer to the origin than for comet E. This implies that comet H travels closer to the sun.

(c) We are given that the sun is at the origin of the coordinate system being considered.

We see that this represents the focus of the elliptic path that comet E takes and the hyperbolic path that comet H takes.

This can be shown by calculating the focus for comet E.

For comet E (ellipse), the focus is:

`(h,k-c)\text{ and }(h,k+c)`

where c is:

`c=√(b^2-a^2)`

For comet E, we will use the focus closer to the sun.

We have that:

`\begin{gathered} c=√(400-144)=√(256) \\ \\ c=16 \end{gathered}`

and the focus is:

`\begin{gathered} (0,-16+16) \\ \\ (0,0) \end{gathered}`

Hence, we see that the focus of the comets is at the origin i.e. the sun.