sos confused ...Ellipse word problem .What's the planet's orbit (distance in millions of miles) about the star?

Question

A planet moves in an elliptical orbit with a star at one of the foci. The perihelion​ (the smallest distance from the planet to the​ star) is

128.03 millions of miles and

the aphelion​ (the largest distance from the planet to the​ star) is

154.93 millions of miles.

Find an equation of the​ planet's orbit around the star.

​[Hint: Set up a coordinate system with the star at one focus

and the major axis lying on the​ x-axis as in the figure. Calculate a from the equation

2a=aphelion + perihelion.

Calculate c from the equation

c=a-perihelion.

Calculate

b=a^2-c^2 Writ equation x^2/a^2 +y^2/b^2 =1

Answer 1

The problem requests that you find the equation of the ellipse in the standard form

(x/a)² + (y/b)² =1 and determine the values for a and b

Let the aphelion = L and the perihelion = S { Large and Small }

Standard equations for the ellipse show that S + L = 2a

The distance from the vertex to a focus is called c and c = sqrt(a² - b²)

For this problem c = a - S which can be rearranged to get L - S = 2 sqrt( a² - b²)

Dividing both sides by L + S = 2a ,and squaring both sides yields ( L -S)²/ (L + S)² = 1 - b²/a²

Substituting for a with a = ( L + S) /2 generates an expression for b, which after a lot of algebra results in

(2 b)² = L² + S² and from before 2a = S + L

From there it is substitution with S = 128.03 and L = 154.93 to get

a = 138.98 and b = 100.49 (million miles)