A CIRCLE is written...

O

OrangeOne

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Sep 8, 2010
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in R^3, ie 3D space as: f(x,y)= x^2+ y^2

If we want to draw level curves we give the function different values of C, for example:
C= 1 --> x^2+ y^2 = 1, This will give us a circle with the center in (0,0) and the radius = sqrt(1)=1

BUT what changes when we have x^2+ 4y^2

What does the 4 change, does it change the centre of the circle?

Please help me!
 
In 3D space,\(\displaystyle z=x^{2}+4y^{2}\) is an Elliptic Paraboloid of the form:

\(\displaystyle z=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\)

I suppose that is \(\displaystyle z=x^{2}+4y^{2}\)?.

Same thing only a=1 and b=1/2.

Here is a rough graph. I done this in the 3D graph from my TI-92.

Glenn's is nicer.
 
\(\displaystyle f(x,y) \ = \ z \ = \ x^2+y^2 \ = \ circular \ paraboloid, \ see \ graph.\)

[attachment=0:3tmfi4c1]eee.jpg[/attachment:3tmfi4c1]
 
\(\displaystyle x^2+y^2 \ = \ 1 \ is \ a \ circle \ of \ radius \ 1, \ but \ x^2+4y^2 \ = \ 1, \ is \ no \ longer \ a \ circle,\)

\(\displaystyle but \ an \ ellipse \ where \ its \ radius \ varies.\)

\(\displaystyle Note: \ I \ suggest \ a \ good \ review \ of \ the \ conics, \ 2d, \ before \ attempting \ 3d.\)
 
Here is a graph in 2D of the ellipse Glenn mentioned.

An ellipse has equation \(\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)

In this case the semi-major axis has length 1 and the semi-minor axis has length 1/2.

See why?.

\(\displaystyle \frac{x^{2}}{1^{2}}+\frac{y^{2}}{(\frac{1}{2})^{2}}=1\Rightarrow x^{2}+4y^{2}=1\)
 
Brilliant last example galactus. I finally get it...

but when i draw level curves of the function f(x,y) = x^2+ 4y^2, i should put the function = constant, so does this mean I should try x^2+ 4y^2= 2, x^2+ 4y^2= 3 etc...even though the equation of an ellips = 1

What would x^2+ 4y^2 be written as if I put x^2+ 4y^2=2

:oops:
 
Yes, it is still an ellipse. Say we have \(\displaystyle x^{2}+4y^{2}=2\). If we divide by the number on the right, in this case 2, we get:

\(\displaystyle \frac{x^{2}}{(\sqrt{2})^{2}}+\frac{y^{2}}{(\frac{1}{\sqrt{2}})^{2}}=1\Rightarrow \frac{x^{2}}{2}+2y^{2}=1\)

an ellipse with semi-major axis length \(\displaystyle \sqrt{2}\) and semi-minor axis length \(\displaystyle \frac{1}{\sqrt{2}}\).

See now?.
 
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