Question

Consider the equation below. x2 − y2 + z2 − 6x − 2y − 2z + 9 = 0 Reduce the equation to one of the standard forms. Classify the surface. ellipsoid hyperboloid of one sheet circular cone elliptic paraboloid hyperboloid of two sheets hyperbolic paraboloid elliptic cylinder parabolic cylinder Sketch the surface.

Answer 1

Answer:

See attachment for detailed answer.

Step-by-step explanation:

Answer 2

The equation of the paraboloid hyperboloid is $\rm (x-3)^2+(y+1)^2+(z-1)^2=0$.

Paraboloid hyperboloid;

The paraboloid is hyperbolic if every other plane section is either a hyperbola or two crossing lines (in the case of a section by a tangent plane).

Given

Equation; $\rm x^2 - y^2 + z^2 -6x-2y - 2z + 9 = 0$

Rewrite the equation;

$\rm x^2 - y^2 + z^2 -6x-2y - 2z + 9 = 0\\\\x^2-6x-y^2-2y+z^2-2z+9=0\\\\(x^2-6x+3^2)-(y^2+2y+1)+(z^2+2z+1)-9+1-1+9=0\\\\(x-3)^2+(y+1)^2+(z-1)^2=0$

The above equation is the surface of an infinite paraboloid.

Hence, the equation of the paraboloid hyperboloid is $\rm (x-3)^2+(y+1)^2+(z-1)^2=0$.

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