Show that for a fixed value of θ, the line parameterized by x=cosθ+t sinθ, y=sinθ-t cosθand z=t

step 1 Okay, so we have the equation of the hyperboloid Now let's substitute the parametric equation in

Show that for a fixed value of θ, the line parameterized by...

Show that for a fixed value of $\theta$, the line parameterized by $x=\cos \theta+t \sin \theta, y=\sin \theta-t \cos \theta$ and $z=t$ lies on the graph of the hyperboloid $x^{2}+y^{2}=z^{2}+1$.

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Key Concepts

Trigonometric Identities

Trigonometric identities such as sin²? + cos²? = 1 are fundamental tools used to simplify expressions that arise when parametric equations involve trigonometric functions. These identities are key in verifying equalities and simplifying the expressions during substitution.

Verification by Substitution

This method consists of plugging the parametric equations into the given implicit equation of a surface to check if the point lies on the surface for all values of the parameter. This technique is essential for proving that a particular parametric representation indeed satisfies the equation defining the geometric object.

Parametric Representation

This concept involves describing curves, surfaces, or lines in space by expressing the coordinates as functions of one or more parameters. Parametrization allows one to analyze and compute properties of geometric objects by turning an implicit relationship into explicit coordinate formulas.

Quadratic Surfaces

Quadratic surfaces are shaped by second-degree equations in three variables. Hyperboloids, which form part of this family, are characterized by equations that mix positive and negative squared terms, creating saddle-like or barrel shapes in space. These surfaces are fundamental in the study of multivariable calculus and differential geometry.

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