# Precalculus

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Mars is in an elliptical orbit with the Sun at one focus, as shown in the figure. At its aphelion, Mars is 156 million miles from the Sun. At its perihelion, Mars in only 128 million miles from the Sun.

How long is the major axis of the ellipse?

Find the minor radius (round to the nearest hundredth).

Find the eccentricity (round to the nearest thousandth).

Write a Cartesian equation.

A planet's equinox (equal day and night) occurs when the angle at the Sun between the major axis and the planet is 90o. How far from the Sun is Mars on the equinox? (Round to the nearest hundredth.)

How far from the Sun are the directrices of the ellipse? (Provide either one of the two directrices rounded to the nearest whole number.)

Write parametric equations for the ellipse.

This looks like Greek to me. -.-

• Precalculus -

You must be studying the ellipse, therefore, must be familiar with the properties of an ellipse such as the a and b values

Can we assume that Mars is at one of the focal points of the elliptical path?
2a = 156+128 = 284
a = 142
c = 142-128 = 14

eccentricity = 14/142 = .0986

also b^2 + c^2 = a^2
b^2 + 14^2 = 142^2
b^2 = 19968
b = √19968 = 141.308

equation:
x^2 / 142^2 + y^2 / 141.308^2 = 1

at equinox, x = 14
14^2/142^2 + y^2/141.308^2 = 1
y^2 / 141.308^2 = .990279706
y^2 = 19773.905
y = 140.62

So it is 140.62 million miles from the sun.

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