Respuesta :
Answer:
1. Plane False
2. Sphere False
3. Ellipsoid False
4. Circular cylinder True
Step-by-step explanation:
For this case we have the following curve [tex]C(t) = (cos t , sin t , t[/tex]
And we can express like this the terms for the curve or each component:
[tex] x= cos t, y= sin t , z =t[/tex]
1. Plane False
The general equation for a plane is given by:
a ( x − x 1 ) + b ( y − y 1 ) + c ( z − z 1 ) = 0.
For this case we don't satisfy this since have sinusoidal functions and this equation is never satisfied.
2. Sphere False
The general equation for a sphere is given by:
(x - a)² + (y - b)² + (z - c)² = r²
And for this case if we see our parametric equation again that is not satisfied since we have two cosenoidal functions. And another function z=t
3. Ellipsoid False
The general equation for an ellipsoid is given by:
x^2/a2 + y^2/b2 + z^2/c2 = 1
And for this case again that's not satisfied since we have
[tex]\frac{cos^2 t}{a^2} + \frac{sin^2 t}{b^2}+\frac{t^2}{c^2} \neq 1[/tex]
4. Circular cylinder True
The general equation for a circular cylinder is given by:
[tex]x^2 +y^2 = r^2[/tex]
And if we replace the equations that we have we got:
[tex] cos^2 t + sin^2 t = 1[/tex] from the fundamental trigonometry property.
So then we see that our function satisfy the condition and is the most appropiate option.
