Thread: Circles and Ellipses

"It’s the year 2742 and you are a spatial technician on the planet of Conicerra. Your job is analyzing air traffic patterns for all space flight around the planet. The latest project is to properly position four ‘traffic lights’ for two new orbital traffic patterns centered around the planet’s capitol city of Conicopolis. One of the patterns is to be circular with radius 4 parsecs and, the other pattern is elliptical with minor axis 6 parsecs and major axis 10 parsecs. Given this information, plot both traffic patterns on a rectangular coordinate system and label the location of all four ‘traffic lights’."

I need help finding the two equations that are in the word problem. I can plot them. Is their a way I can plot them on this site?

Hello, JohnMon!

It’s the year 2742 and you are a spatial technician on the planet of Conicerra.
Your job is analyzing air traffic patterns for all space flight around the planet.
The latest project is to properly position four ‘traffic lights’ for two new orbital traffic patterns
centered around the planet’s capitol city of Conicopolis.

One of the patterns is to be circular with radius 4 parsecs and
the other pattern is elliptical with minor axis 6 parsecs and major axis 10 parsecs.

Given this information, plot both traffic patterns on a rectangular coordinate system
and label the location of all four traffic lights.

Of course, we'll place Conicopolis at the origin.
And I assume the traffic lights are placed at the intersections of the two orbits.

The equation of the circular orbit is: .$\displaystyle x^2 + y^2 \:=\:4^2 \quad\Rightarrow\quad y^2 \:=\:16-x^2\;\;{\color{blue}[1]}$

The equation of the elliptic orbit is: .$\displaystyle \frac{x^2}{5^2} + \frac{y^2}{3^2} \:=\:1 \quad\Rightarrow\quad 9x^2 + 25y^2 \:=\:225\;\;{\color{blue}[2]}$

Substitute [1] into [2]: .$\displaystyle 9x^2 + 25(16-x^2) \:=\:225 \quad\Rightarrow\quad x^2 \:=\:\frac{175}{16} \quad\Rightarrow\quad x \:=\:\pm\frac{5\sqrt{7}}{4}$

Substitute into [1]: .$\displaystyle y^2 \:=\:16 - \frac{175}{16} \:=\:\frac{81}{16}\quad\Rightarrow\quad y \:=\:\pm\frac{9}{4}$


The four traffic lights are located at: .$\displaystyle \left(\pm\frac{5\sqrt{7}}{4},\;\pm\frac{9}{4}\righ t) $