Question

Question 9: For the ellipse with a = 20 AU and e = 0.5, can you find a point in the orbit where r1 and r2 are equal? Sketch the ellipse, the location of this point, and r1 and r2 in the space below.

Question 10: What is the value of the sum of r1 and r2 and how does it relate to the ellipse properties? Is this true for all ellipses?

Answer 1

Question 9)

The properties of the ellipse are given as

the length of semi-major axis, a = 20 AU

Astronomical Unit ( AU ) is the average distance between the earth and the sun. = 150 million Kilometers

Eccentricity,   

Now, C = distance between any one of the focal points, F₁ or F₂ with the center of the ellipse.

Now from the figure,

since the distance between the focal point to the center of the ellipse is the same, the radius R₁ and R₂ will be equal when the point ( X , Y ) is at the vertical axis of the ellipse.

i.e

By the equation of the ellipse,

We get

Question 10 )

Let R1 be the distance from the focus at (-c,0) to the point at (x,y). Since this is the distance between two points, we'll need to use the distance formula.

Similarly, R2 will involve the distance formula and will be the distance from the focus at the (c,0) to the point at (x,y)

We can use the fact that the vertices are on the ellipse to find out what the sum of the distances is

i.e.

If we take the vertex on the right, then

R1 + R2 = ( a + c ) + ( a - c ) = 2 a

Therefore, the constant is 2a and R1 + R2 = 2 a for every point on an ellipse. This equation is true for all ellipses.

When R1 = R2 = R ( say ), then R = a

Thus R = 20 AU

From the earlier figure, consider a right andled triangle with sides R1, the positive Y-axis, and the negative X-axis.

Then,

Then the point on the orbit when the two radii are equal is (0, b ) =   

Hope This Helps

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