Answer to Question #90263 in Analytic Geometry for Saudagar

Question #90263
Check whether the surface represented by
x^2 + y^2 = 2 (z + 1) is symmetric about the YZ-plane, ZX-plane and XY-plane. Do the coordinate axes intersect the surface?
1
Expert's answer
2019-05-27T13:05:13-0400

Solution. Considering this equation, we see that the surface x^2 + y^2 = 2 (z + 1) is an elliptic paraboloid. Find the intersection of the surface of the planes parallel to the XY-plane (z=h, where h is constant). Get


"x^2+y^2=2(h+1)"

For h=-1 get point (0,0,-1); for h>-1 get circle x^2+y^2=2(h+1). For XY-plane (h=0) get x^2+y^2=2. Therefore the surface intersects the coordinate axes 0x and 0y at the points:


"0x: (-\\sqrt2, 0,0) and (\\sqrt2, 0,0)"

"0y: (0,-\\sqrt2,0) and (0, \\sqrt2,0)"

The surface relative to the XY-plane is not symmetrical.

Find the intersection of the surface of the planes parallel to the YZ-plane (x=h, where h is constant). Get


"y^2=2z+2-h^2."

This equation is a parabola equation. For YZ-plane (x=0) get y^2 = 2 (z + 1). The top of the parabola at the point (0,0,-1). Parabola intersects the coordinate axis 0y at points:


"0y: (0,-\\sqrt2,0) and (0, \\sqrt2,0)"

Considering the symmetric planes x = -h and x = h, we get the same parabolas y^2 = 2 (z + 1)-h^2. Hence the surface relative to the YZ-plane is symmetrical.

Find the intersection of the surface of the planes parallel to the ZX-plane (y=h, where h is constant). Get


"x^2=2z+2-h^2."

This equation is a parabola equation. For ZX-plane (y=0) get x^2 = 2 (z + 1). The top of the parabola at the point (0,0,-1). Parabola intersects the coordinate axis 0y at points:


"0x: (-\\sqrt2, 0,0) and (\\sqrt2, 0,0)"

Considering the symmetric planes y = -h and y = h, we get the same parabolas x^2 = 2 (z + 1)-h^2. Hence the surface relative to the ZX-plane is symmetrical.

Answer. The surface is symmetric about YZ-plane and ZX-plane. The surface relative to the XY-plane is not symmetrical.

The surface intersects the coordinate axes in points:


"0x: (-\\sqrt2, 0,0) and (\\sqrt2, 0,0)"

"0y: (0,-\\sqrt2,0) and (0, \\sqrt2,0)"

"0z: (0,0,-1)"


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