# Quadratic surfaces standard form help

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1. Jan 31, 2017

### Mark53

1. The problem statement, all variables and given/known data

Suppose a quadratic equation in 3 variables is put into a standard form represents a hyperboloid of one sheet. This hyperboloid has the property that:

• the cross section through z= 0 is a circle of radius 1;

• the cross section through x= 1 is the two straight lines given (in the plane x = 1) by y = ± (1/2) z

What is the standard form and the original equation in quadratic form?

2. Relevant equations

hyperboloid of one sheet is given by

x^2/a^2+y^2/b^2-z^2/c^2=1

3. The attempt at a solution

using z=0

x^2/a^2+y^2/b^2=1 which is an ellipse

Given that it is a circle it means that b=a which means that it equals 1 as that is what the radius is.

x^2+y^2-z^2/c^2=1

How would I find C from here

Last edited: Jan 31, 2017
2. Jan 31, 2017

### haruspex

Use the other piece of information:

3. Jan 31, 2017

### Mark53

1+(1/4)z^2-z^2/c^2=1

(1/4)z^2-z^2/c^2=0

z^2((1/4)-c^2)=0

(1/4)-c^2)=0

c^2=1/4

c=1/2

therefore

x^2+y^2-z^2/(1/2)^2=1

4. Jan 31, 2017

### pasmith

Why not simplify 1/(1/2)^2 as 4?

5. Jan 31, 2017

### haruspex

Check that step.

6. Jan 31, 2017

### Mark53

that would mean that c=2

7. Jan 31, 2017

Yes.