Deriving Standard Form of Ellipse Equation

  • Thread starter Mahmoud2010
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In summary, in the derivation of the standard form of the equation of an ellipse, we squared both sides of the original equation, which could potentially produce a candidate that does not satisfy the original equation. However, all steps in the derivation are reversible and the only potential issue arises when taking square roots, but since both sides are positive numbers, there is no possibility for the solution to be ±b. The speaker also expresses their own understanding and gratitude for the help provided.
  • #1
Mahmoud2010
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in the derivation of the standard form of equation of ellipse

[PLAIN]http://img694.imageshack.us/img694/8324/capturedpw.jpg

we squared both sides of equation isn't that means that we have produce we have produced a candidate which doesn't satisfy the original equation.

Thanks
 
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  • #2
There is no problem because all of your steps are reversible. When you work the steps in reverse order, the only place where there is a potential problem is where you take square roots. But if you look carefully, you will see that you are only taking the positive square root of both sides, which themselves are positive numbers. And if you have two positive numbers a and b with a2=b2, then a = b. You don't have the possibility that a = ±b.
 
  • #3
I think of this also . but I want to be more certain.

At all Thanks for help.
 

FAQ: Deriving Standard Form of Ellipse Equation

1. What is the standard form of an ellipse equation?

The standard form of an ellipse equation is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) represents the center of the ellipse and a and b are the lengths of the major and minor axes, respectively.

2. How do you derive the standard form of an ellipse equation?

To derive the standard form of an ellipse equation, you must first have the equation of an ellipse in general form: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. Then, you must complete the square for both the x and y terms, and finally, factor out a common factor to get the equation in the standard form.

3. What is the significance of the values a and b in the standard form of an ellipse equation?

The values a and b represent the lengths of the major and minor axes of the ellipse, respectively. The major axis is the longest diameter of the ellipse and passes through the center, while the minor axis is the shortest diameter and also passes through the center.

4. Can the standard form of an ellipse equation be used for all ellipses?

Yes, the standard form of an ellipse equation can be used for all ellipses, regardless of their orientation or size. It is the most common and convenient form for representing ellipses in mathematics and science.

5. How is the standard form of an ellipse equation related to the eccentricity of an ellipse?

The eccentricity of an ellipse is defined as the ratio of the distance between the foci to the length of the major axis. The standard form of an ellipse equation can be used to calculate the eccentricity, as e = √(1 - b^2/a^2). This means that the larger the value of a in the standard form, the smaller the eccentricity of the ellipse.

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