Answer:
[tex]x=1\,,\,y=t\,,\,z=t[/tex]
Step-by-step explanation:
A line is a one-dimensional figure that has no thickness and extends infinitely in both directions.
A line segment is a line that has two end-points and a ray is a line that has one end-point.
Given equation is [tex]x^2+y^2-z^2=1[/tex]
To find : a line that lies entirely in the set defined by the given equation.
Solution:
Take [tex]x=1\,,\,y=t\,,\,z=t[/tex]
Check:
[tex]L.H.S=1^2+t^2-t^2=1+0=1=R.H.S[/tex]
Therefore, [tex]x=1\,,\,y=t\,,\,z=t[/tex] satisfy the given equation.
We want to find a line that lies in the set defined by x^2 + y^2 - z^2 = 1
That line can be <1, t, t>.
So we want to have a line that is defined by the given restriction.
We can rewrite the restriction as:
x^2 + (y^2 - z^2) = 1
Now we can add another restriction (just to find the line) that is defining:
y = z (notice that if we fix the value of x, this will define a line).
Then we will have:
x^2 + (y^2 - y^2) = 1
x^2 = 1
Now we can fix a value for x, for example x = 1
Then our line is defined by:
y = z and x = 1.
Writing it in vector form we would have something like:
<1, t, t>
Replacing that in the restriction, we get
1^2 + t^2 - t^2 = 1 + 0 = 1
As expected.
If you want to learn more, you can read:
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