Finding the equation of an ellipse from foci and directrices

Thread starter kasse
Start date Jan 13, 2007
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Ellipse

In summary, an ellipse is a geometric shape that has two foci and two directrices, which are essential in determining its equation. The standard form of the equation is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes. Real-life applications of finding the equation of an ellipse include architectural design, satellite orbits, and predicting celestial orbits. There are various methods and shortcuts to find the equation, such as using the distance formula or utilizing the properties of ellipses.

Jan 13, 2007

kasse

How can one find the eq. of an ellipse given that the foci are (-2,0) and (2,0) and that the directrices are x=-8 and x=8?

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Jan 13, 2007

cristo

Staff Emeritus

Science Advisor

This seems like a homework question. What have you tried?

Jan 13, 2007

berkeman

Mentor

Moved to precalc homework.

Kasse, you need to show your own work in order to get our help. What are the equations that define an ellipse?

1. What is an ellipse and how is it different from a circle?

An ellipse is a geometric shape that resembles a stretched circle. It is defined as the locus of all points in a plane, the sum of whose distances from two fixed points (called the foci) is constant. Unlike a circle, which has a constant radius, an ellipse has two different radii (major and minor) and is not perfectly symmetrical.

2. How do the foci and directrices relate to the equation of an ellipse?

The foci and directrices are key elements in determining the equation of an ellipse. The foci are two points inside the ellipse, while the directrices are two lines perpendicular to the major axis of the ellipse. The distance between the foci and the sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis. The directrices help determine the eccentricity of the ellipse, which is a key parameter in the equation.

3. Can you explain the standard form of the equation of an ellipse?

The standard form of the equation of an ellipse is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse, a is the length of the semi-major axis, and b is the length of the semi-minor axis. The values of a and b can be determined using the distances between the foci and the directrices.

4. What are some real-life applications of finding the equation of an ellipse?

Ellipses are commonly found in the design of architectural structures, such as arches and domes. They are also used in the design of satellite orbits and in predicting the orbits of celestial bodies. In addition, ellipses are used in engineering and physics to model the motion of objects under the influence of gravitational forces.

5. Are there any shortcuts or formulas for finding the equation of an ellipse?

Yes, there are several formulas and shortcuts that can be used to find the equation of an ellipse. One method is to use the distance formula to determine the distances between the foci and the directrices, and then use these values to solve for a and b in the standard form of the equation. Another method is to use the properties of ellipses, such as the eccentricity and the focus-directrix relationship, to derive a simplified equation. Additionally, there are online calculators and software programs that can quickly generate the equation of an ellipse given the foci and directrices.