Question #114939
Trace the conicoid represented by
x^2 +2z^2 = y. Also describe its sections by the planes x = c,∀c ∈ R.
1
Expert's answer
2020-05-18T18:50:06-0400

x2+2z2=yx^2+2z^2=y

Let's rewrite equation: 2x2+4z2=2y    x20.5+z20.25=2y2x^2+4z^2=2y\iff \frac{x^2}{0.5}+\frac{z^2}{0.25}=2y

Now we can see that our surface is elliptic paraboloid.


Denominators are unequal. It means that our paraboloid is not a paraboloid of revolution.


Let x=0x=0, then y=2z2y=2z^2

And let z=0z=0, then y=x2y=x^2

y=2z2y=2z^2 and y=x2y=x^2 principal parabolas.



Let's look on sections.

y=cy=c, where cRc\isin R

x2+2y2=cx^2+2y^2=c - ellipses if c>0c>0



If x=c,x=c, where cRc\in R

y=2z2+c2y=2z^2+c^2 - parabolas.

c2c^2 - is distance between the origin and vertex of parabola.



A similar result will be if we look at z=c,z=c, where cRc\in R

y=x2+2c2y=x^2+2c^2



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