I don't even how to go about this....

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?

What is the major radius?

Find the focal radius.

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. -.-

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
Calculate major radius etc, as defined in your text.

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.

I understand that this may seem overwhelming at first, but I'll break it down step-by-step to help you understand and solve the problem.

1. To find the length of the major axis of the ellipse, we need to consider the distances at aphelion and perihelion. The major axis of an ellipse is defined as the longest chord, which passes through the center and is perpendicular to the minor axis. In this case, the major axis represents the longest distance between Mars and the Sun. So, the length of the major axis is the difference between the distances at aphelion and perihelion:
Major axis length = Aphelion distance - Perihelion distance

2. The major radius of the ellipse is half the length of the major axis. So, to find the major radius, you divide the major axis length by 2:
Major radius = Major axis length / 2

3. The focal radius is the distance between the center of the ellipse and either of the two foci. The foci of an ellipse lie on the major axis and are equidistant from the center. To find the focal radius, you take half the difference between the distances at aphelion and perihelion:
Focal radius = (Aphelion distance - Perihelion distance) / 2

4. The minor radius of the ellipse is half the length of the minor axis. The minor axis is perpendicular to the major axis and represents the shortest distance between the two sides of the ellipse. To find the minor radius, you need to use the formula for the semi-minor axis:
Minor radius = √ [(Major radius)^2 - (Focal radius)^2]

5. The eccentricity of an ellipse is a measure of how elongated or stretched it is. It can be calculated using the formula:
Eccentricity = (Focal radius) / (Major radius)

6. The Cartesian equation of an ellipse centered at the origin is given by:
x² / a² + y² / b² = 1, where 'a' represents the semi-major axis length, and 'b' represents the semi-minor axis length of the ellipse.

7. To find how far Mars is from the Sun on the equinox, when the angle at the Sun between the major axis and the planet is 90 degrees, you can use the Pythagorean theorem. Let 'd' represent the distance from Mars to the Sun on the equinox. Then:
d² = (Major radius)² + (Focal radius)²

8. The directrices of an ellipse are two imaginary lines that are parallel to the minor axis. They are equidistant from the center and perpendicular to the major axis. To find the distance from the Sun to the directrices, you can use the formula:
Distance from the Sun to the directrix = Major radius / Eccentricity

9. The parametric equations of an ellipse can be written in terms of the angles 'θ' and the semi-major and semi-minor axes length 'a' and 'b' respectively:
x = a cos(θ)
y = b sin(θ)

I hope this step-by-step breakdown helps you understand how to solve the given problem involving Mars' elliptical orbit around the Sun.

Don't worry! I'm here to help you break it down step by step.

1. To find the length of the major axis of the ellipse, you need to find the distance between the two farthest points of the ellipse. In this case, the distance between the aphelion and perihelion of Mars. The major axis is the longest diameter of the ellipse. So, you can subtract the distance at the perihelion from the distance at the aphelion to obtain the length of the major axis.

Major Axis = Distance at Aphelion - Distance at Perihelion

2. The major radius is half the major axis. You can divide the length of the major axis by 2 to find the major radius.

Major Radius = (Length of Major Axis) / 2

3. The focal radius is the distance between the center of the ellipse to one of its foci. To find the focal radius, you can subtract the distance at the perihelion from the distance at the aphelion and divide it by 2.

Focal Radius = (Distance at Aphelion - Distance at Perihelion) / 2

4. The minor radius is half the difference between the major axis and the focal radius. You can subtract the focal radius from the major radius and divide it by 2.

Minor Radius = (Major Radius - Focal Radius) / 2

5. The eccentricity of the ellipse can be found using the formula:

Eccentricity = (Focal Radius) / (Major Radius)

6. To write a Cartesian equation of the ellipse, you can use the standard form for an ellipse:

((x - h)^2) / (a^2) + ((y - k)^2) / (b^2) = 1

Here, (h, k) represents the center of the ellipse, a is the semi-major axis, and b is the semi-minor axis.

7. On the equinox, when the angle between the major axis and the planet is 90 degrees, Mars will be at its greatest distance from the Sun. Therefore, the distance from the Sun on the equinox is equal to the semi-major axis.

Distance from the Sun on the Equinox = Semi-Major Axis

8. The directrices of the ellipse are given by the formula:

Directrices = Semi-Major Axis / Eccentricity

You can choose either one of the two directrices and round it to the nearest whole number.

9. Finally, to write parametric equations for the ellipse, you can use the following formulas:

x = h + a * cos(t)
y = k + b * sin(t)

Here, (h, k) represents the center of the ellipse, a is the semi-major axis, b is the semi-minor axis, and t represents the parameter that ranges from 0 to 2π.

I hope this breakdown helps you understand how to solve the given problem! Let me know if you have any more questions.