Answer to Question #200497 in Analytic Geometry for tanya

Question

Identify and trace the conicoid y² +z² = x. Describe its sections by the planes x = 0, y = 0 and z = 0

Answer 1

$y^2+z^2=x$

Now we can see that our surface is spherical ellipsoid.

Denominators are equal. It means that our ellipsoid is not a ellipsoid of revolution.

Let $y=0$, then $z^2 = x$

And let $z=0$, then $y^2=x$

$z^2 = x$ and $y^2=x$ principal parabolas.

Let's look on sections.

$\implies$ $y=0$

$\implies$ $z^2 = x$ ...................Parabola with its axis along x - axis.

$\implies$  $z=0$

$\implies$$y^2 = x$ .................Parabola with its axis along x - axis.

$\implies$ $x=0$ 

$\implies$$y^2 + z^2 = 0$ 

$y^2+z^2=x$

Now we can see that our surface is spherical ellipsoid.

Denominators are equal. It means that our ellipsoid is not a ellipsoid of revolution.

Let $y=0$, then $z^2 = x$

And let $z=0$, then $y^2=x$

$z^2 = x$ and $y^2=x$ principal parabolas.

Let's look on sections.

$\implies$ $y=0$

$\implies$ $z^2 = x$ ...................Parabola with its axis along x - axis.

$\implies$  $z=0$

$\implies$$y^2 = x$ .................Parabola with its axis along x - axis.

$\implies$ $x=0$ 

$\implies$$y^2 + z^2 = 0$ ...........It represents x- axis.

Thecurve is shown below: