Question

what surface is represented by x^2 + y^2 = 9z ? Give a rough sketch of it. Obtain the section
of this surface by the plane y = 0.

Answer 1

Answer on Question #67791 – Math – Analytic Geometry

Question

What surface is represented by $x^2 + y^2 = 9z$?

Give a rough sketch of it. Obtain the section of this surface by the plane $y = 0$.

Solution

Elliptic paraboloid

$$\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$$

We have that

$$\frac{x^2}{9} + \frac{y^2}{9} = z$$ $$c = 1, a^2 = b^2 = 9$$

Therefore, there is circular paraboloid.

The trace, or cross section, in the $xy$-plane is a point.

If $c = 1$, the point is the origin $(0,0)$.

The traces in planes parallel to and above the $xy$-plane are circles.

The traces in the $yz$-plane and $xz$-plane are parabolas, as the traces are in planes parallel to these.

The cross section of the surface $x^2 + y^2 = 9z$ by the plane $y = 0$ is the parabola $z = \frac{x^2}{9}$ lying in the plane $y = 0$. It opens up.

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