(https://www.uwgb.edu/dutchs/MATHALGO/Ellipses.HTM)
(http://en.m.wikipedia.org/wiki/Ellipse)
The formula for an ellipse is (x-h)2 /b2 + (y-k)2 /a2=1, if it is "skinny"; or (x-h)2 /a2 + (y-k)2 /b2=1, if it is fat. You identify an ellipse by noticing that: both x and y are squared, the equation adds both x and y terms on one side, and they have different coefficients.To get the equation of an ellipse, first group the Xs and Ys and move the constant to the other side of the "equals" sign; next, take out the GCF of X and put it out side the parenthesis (same procedure for Y) and put the GCF of both on the other side of the "equals" sign well; then you must complete the square for x and y; then, factor both perfect square trinomials and simplify the opposite side; finally, divide the "x and y" part of the equation by what it is equal to to get the equation to equal 1, and reduce the fractions.
An ellipse looks like a stretched circle because a circle's eccentricity is 0, but an ellipses' eccentricity is between 0 and 1. It is much easier to get the key points of an ellipse when it is in standard form. All ellipses consist of: being shaped either like a "skinny" or "fat' circle, a center, two vertices, two co-vertices, it's two foci, and it's eccentricity. If the bigger number in the equation lies under the x it will be fat, but if it is under y it will be skinny. The center of an ellipse is (h,k). The major axis is horizontal if the bigger number is under the x term and if its under the y term it is vertical. The major axis connects the two vertices together (length of 2a). The minor axis connects the two co-vertices together (length of 2b). The foci determines how stretched out the ellipse is, if the focus is close to zero it makes the ellipse more circularly shaped, but if it is closer to 1, it stretches out more. To find a missing value such as a, b, or c; use the equation c2 =a2 -b2--CAUTION: this formula will only work if you have two of the mentioned values.
Ellipses are used in the real world, like our solar system. Johannes Kepler discovered that each planet travels around the sun in an elliptical orbit with the sun being the focus. This also applies to an atom, with the electrons orbiting elliptically and the nucleus being the focus. In Lithotripsy, a medical procedure for treating kidney stones, the patient is placed in an elliptical tank of water, the kidney stone at one focus and high-energy shock waves generated at the other focus are concentrated on the stone, to destroy it.
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