Respuesta :
Answer:
Circular paraboloid
Step-by-step explanation:
Given ,
[tex]r(s,t)=ssin4t,s^2,scos4t[/tex]
Here, these are the respective [tex]x,y,z[/tex] axes components.
- Component along x axis [tex]r_i : ssin4t[/tex]
- Component along y axis [tex]r_j : s^2[/tex]
- Component along z axis [tex]r_k : scos4t[/tex]
We see that , from the parameterised equation , [tex]r_i^2+r_k^2=s^2sin^24t+s^2cos^24t\\r_i^2+r_k^2=s^2\\r_i^2+r_k^2=r_j[/tex]
This can also be written as :
[tex]x^2+z^2=y[/tex]
This is similar to an equation of a parabola in 1 Dimension.
By fixing the value of z=0,
We get [tex]y=x^2[/tex] which is equation of a parabola curving towards the positive infinity of y-axis and in the x-y plane.
By fixing the value of x=0,
We get [tex]y=z^2[/tex] which is equation of a parabola curving towards positive infinity of y-axis and in the y-z plane.
Thus by fixing the values of x and z alternatively , we get a CIRCULAR PARABOLOID.
Following are the calculation to the given equation:
Given vector equation:
[tex]\to r(s, t)= (s \sin 4t,s^2, s \cos 4t)[/tex]
The corresponding parametric equations for the given surface is given by,
[tex]x=s \sin 4t\\\\ y =s^2\\\\ z=s \cos 4t[/tex]
For any point (x,y,z) we have that,
[tex]x^2+z^2 = (s \sin 4t)^2 +(s \cos 4t)^2[/tex]
[tex]= s^2 \sin^2 4t +s^2 \cos^2 4t \\\\= s^2 (\sin^2 4t+cos^2 4t) \\\\= s^2 (1) \\\\=s^2\\\\=y \\[/tex]
So, after eliminating the parameters, we get, [tex]x^2 +z^2 = y[/tex]
Therefore, the answer is "Option c".
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