Question

the elliptical orbit of Mars each unit of the coordinate plane represents 1 million kilometers. The planet's maximum distance from the sun is 249 million kilometers and its minimum distance from the Sun is 207 million kilometers. The Sun is at one focus of the ellipse and the center of the ellipse is at (0, 0). The coordinates of the Sun are (?, 0)

Answer 1

The sun has the same y coordinates as the center so it implicates it is horizontal ellipse. so use the formula for horiz ellipse,\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\] the max distance from the sun is 249,000,000Km so \[a+c=249*10^{6}km\] and the minimum is 207,000,000km so
\[a-c=207*10^{6}km\] add the two equations to eliminate c terms. \[2a=249*10^{6}km+207*10^{6}km = 456*10^{6}km\]divide by 2. and you get \[\frac{456*10^{6}}{2}=228*10^{6}km=a\] substitute a in the first equation. \[(228*10^{6}km)+c=249*10^{6}km\] subtract \[c=249*10^{6}km-228*10^{6}km = 21*10^{6}km\] so the sun is at \[(21*10^{6},0)\] or (21,000,000,0)

Answer 2

forgot to mention the c,c is the Foci (+-c,0), where \[c^{2}=a^{2}+b^{2}\]

Answer 3

thanks!