Question

Pre-Calculus Conic SectionsPart 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

A) (0,4) and (0,-36)

B) Comet H

C) The sun is one of the two foci of that Ellipse and Hyperbole

A) We need to plot those hyperboles:

`\begin{gathered} E)((y+16)^2)/(400)+(x^2)/(144)=1 \\ H)((y+13)^2)/(144)-(x^2)/(25)=1 \end{gathered}`

And solve this geometrically E:

Note that the vertices are the points that intercept the "greater axis, so we have vertices on (0,4) and (0,-36)

B) Now let's place the other conic section, the B one. So let's compare them:

Since the sun is located at the origin (0,0) then Comet H travels closer to the sun, according to the graph above.

C) Note that the key features of an ellipse are focus and directrix. The sun is in the Origin.

For Comet E, the Sun is one of the focus:

`\begin{gathered} \mleft(h,\: k+c\mright),\: \mleft(h,\: k-c\mright) \\ \mleft(0,\: -16+c\mright),\: \mleft(0,\: -16-c\mright) \\ \mleft(0,\: 0\mright),\: \mleft(0,\: -32\mright) \end{gathered}`

Considering that h=0, k= -16 is the center of that ellipse. (0,-16) For the Hyperbole we have:

`\begin{gathered} Focus=\mleft(h,\: k+c\mright),\: \mleft(h,\: k-c\mright) \\ c=\sqrt[]{a^2+b^2} \\ c=\sqrt[]{12^2+5^2}=13 \\ \mleft(0,\: -13+13\mright),\: \mleft(0,\: -13-13\mright) \\ (0,0),(0,-26) \end{gathered}`

So the sun is one of those two focus of that Hyperbole.

Hence, the answers are

A) (0,4) and (0,-36)

B) Comet H

C) The sun is one of the two foci of that Ellipse and Hyperbole