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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

How to solve this problem

3(x^2+y^2)+2xy=88.

Some solutions do the first derivative and then substitute back into the equation, but why?

Can we solve it purely by algebra?

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi;

There is an algebraic solution and it yields

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

Thanks bobby, but why does the first derivative method work here and not in the following sum? (7x-2)^(1/3)+(7x+5)^(1/3)=3.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

First, do you see the algebraic way?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

can you help me with the algebraic way?

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

3(x^2+y^2)+2xy=88

Since at a root the equation crosses the x axis that means y = 0. So set y = 0 and you get

3x^2 = 88

Can you solve that for x or do you need some help?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

Ok so x is +/- 2*((22/3)^(1/2)).

How do we know that it crosses the x axis?

Is this the equation of an ellipse?

*Last edited by NakulG (2016-01-14 15:43:12)*

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

It pretty much is the definition of a root or a zero of a function.

Wikipedia wrote:

If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis. An alternative name for such a point (x,0) in this context is an x-intercept.

It does look like an ellipse, the red points are the roots.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

What would be the equation if the ellipse is not cutting any axis?

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

There should be an infinite number of such ellipses. The roots of the equation would then be complex numbers.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

Ok, can you point me to some online / resource where I can read and practice more on the conic section.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Start here:

http://www.mathsisfun.com/geometry/conic-sections.html

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

Thanks, I will go thru these.

In the question that I had mentioned, what is the significance of the xy term in the equation? Can you suggest.

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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

The significance of the xy term is that it tells about the rotation of the figure (ellipse), in this case as the xy is not equal to zero it is (the ellipse) is not parallel to either axis, x or and y.

*Last edited by NakulG (2016-01-15 04:37:50)*

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The equation

represents an ellipse tilted at an angle of 45° to the coordinate axes. The *xy* term can be eliminated by making the substitution

and working in the *uv*-plane.

NOTE that for this particular problem you don't need to make this substitution because you can simply put *y* = 0.

*Last edited by Nehushtan (2016-01-16 19:47:04)*

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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

This is a good resource for rotation of axis in conic section ... http://math.sci.ccny.cuny.edu/document/show/2685

*Last edited by NakulG (2016-01-15 22:56:06)*

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