Answer :
Solution :
Let the center be (0,0)
Now from the diagram given below ,
The major axis = periapsis + apoapsis
2 x semi major axis = periapsis + 98
2 x 93 = 98 + periapsis
Therefore, periapsis = 186 - 98
= 88 million miles
Star coordinate = (93-98, 0)
= (-5,0)
Let the minor axis be 2b
For ellipse [tex]$c^2 = a^2 - b^2$[/tex]
[tex]$(-5)^2 = (-93)^2 - b^2$[/tex]
[tex]$b^2=8624$[/tex]
[tex]$b=\sqrt{8624}$[/tex]
Now, equation of the ellipse
[tex]$\frac{x^2}{(93)^2}+\frac{y^2}{(\sqrt{8624})^2}=1$[/tex]
[tex]$\frac{x^2}{(93)^2}+\frac{y^2}{8624}=1$[/tex]
The equation for the orbit of the planet around the star is;
(x²/93²) + (y²/(√8624)²) = 1
The length of the periapsis of the given planet is;
88 million miles
We are told the orbit around its' star is an ellipse.
We are told that apoapsis of a planet is its greatest distance from the star, and the periapsis is its shortest distance. Thus;
major axis = periapsis + apoapsis
Now, we are told that; the apoapsis of the planet 98 million miles.
Also, that The mean distance from a planet to its star is 93 million miles.
Thus, major axis/2 = 93
Thus, major axis = 93 × 2
Major axis = 186 million miles
Thus;
186 = periapsis + 98
periapsis = 186 - 98
periapsis = 88 million miles
To get the equation of the orbit of the planet around the star, lets first find the focus coordinate.
We know it's y-coordinate will be zero since its the focus but it's x-coordinate will be; 93 - 98 = -5
Thus; star coordinate = (-5, 0)
As we are dealing with ellipse, we have gotten;
a = 93 and c = -5
Thus; b = √(a² - c²)
b = √(93² - (-5)²)
b = √8624
Formula for equation of the ellipse is;
(x²/a²) + (y²/b²) = 1
Thus, we have;
(x²/93²) + (y²/(√8624)²) = 1
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