Consider the curve C with parametrisationwhere t E R.r(t) ==cos(5t)cosh(t)i+sin(5t)cosh(t)-j+tanh(t)k,(a) Show that C lies on the surface of the sphere described by the equation x² + y² + z² = 1.(b) Find the vector equation of the line L that is tangent to C at the point r(0).

The general form of a curve in three dimensions is r(t)=f(t)i+g(t)j+h(t)k, where f, g and h are the…

Answered: Consider the curve C with…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider the curve C with parametrisation
where t E R.
r(t) =
=
cos(5t)
cosh(t)
i+
sin(5t)
cosh(t)
-j+tanh(t)k,
(a) Show that C lies on the surface of the sphere described by the equation x² + y² + z² = 1.
(b) Find the vector equation of the line L that is tangent to C at the point r(0).

Expert Solution

Step 1

The general form of a curve in three dimensions is $\begin{array}{rcl} r (t) & = & f (t) i + g (t) j + h (t) k \end{array}$ , where f, g and h are the functions of the parameter t. In this problem, the given curve is $\begin{array}{rcl} r (t) & = & \frac{\cos (5 t)}{\cosh (t)} i + \frac{\sin (5 t)}{\cosh (t)} j + \tanh (t) k \end{array}$ .

In the first part of the problem, we have to show that the curve lies on the surface of a sphere $x^{2} + y^{2} + z^{2} = 1$ . In the second part of the problem, we have to find the vector equation of the line tangent line at the point $t = 0$ .