For each of the following equations, select the item in the list below that corr

Question

For each of the following equations, select the item in the list below that corresponds to the graph of the given equation in three-dimensional space. 7x^2 - 2y^2 + 4z^2 = 3z^2 + 5x^2 9x^2 - 8z^2 - 4 = 6x^2 - 9z^2 - 2y^2 7y^2 = 5x - 2z^2 8x^2 - 3 = -y^2 - 2 5y^2 + 9x^2 + 3 = 8x^2 - 4z^2 + 7y^2 List of Surfaces Sphere Ellipsoid but not a sphere Hyperboloid of one sheet Hyperboloid of two sheets Circular cylinder Cone Elliptic but not circular paraboloid Circular Paraboloid Plane Hyperbolic Paraboloid Hyperbolic Cylinder Elliptical Cylinder

Explanation / Answer

a) 7x^2 - 2y^2 + 4z^2 = 3z^2 + 5x^2

Rearranging :
2x^2 - 2y^2 + z^2 = 0

2x^2 + z^2 = 2y^2

This has x,y,z all...
Two positive and one negative....

Since it is of the form x^2/a^2 + z^2/c^2 = y^2/b^2,
this is a CONE opening along the y-axis

CONE(option C)

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b)
9x^2 - 8z^2 - 4 = 6x^2 - 9z^2 - 2y^2

Rearrange :
3x^2 + z^2 + 2y^2 = 4

Divide all over by 4 :
3x^2/4 + y^2/2 + z^2/4 = 1

This is of the form
x^2/a^2 + y^2/b^2 + z^2/c^2 = 1

So, this is an ELLIPSOID(option B)

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c) 7y^2 = 5x - 2z^2

Rearrange :
7y^2 + 2z^2 = 5x

This is of the form
y^2/b^2 + z^2/c^2 = x/a

So, this is an ELLIPTIC PARABOLOID(option G)

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d) 8x^2 - 3 = -y^2 - 2

Rearrange :
8x^2 + y^2 = -2 + 3

8x^2 + y^2 = 1

This is of the form x^2/a^2 + y^2/b^2 = 1
Notice the x and y terms do not have the same coefficient...

So, ELLIPTIC CYLINDER(option L)

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e)
5y^2 + 9x^2 + 3 = 8x^2 - 4z^2 + 7y^2

x^2 + 4z^2 - 2y^2 + 3 = 0

-x^2 - 4z^2 + 2y^2 = 3

Divide all over by 3 :
-x^2/3 - 4z^2/3 + 2y^2/3 = 1

This is a HYPERBOLOID OF TWO SHEETS(option D)

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