   Chapter 13.1, Problem 43E

Chapter
Section
Textbook Problem

# Find a vector function that represents the curve of intersection of the two surfaces.43.  The cone z  =  x 2  +  y 2 and the plane z = 1 + y

To determine

To find: The vector function of curve that intersects two surfaces.

Explanation

Given data:

The equations of cone is z=x2+y2 and plane is z=1+y .

Definition:

Consider parametric equations as x=f(t) , y=g(t) , and z=h(t) .

Write the expression for vector function in terms of parametric equations.

r(t)=f(t)i+g(t)j+h(t)k (1)

Here,

f(t) , g(t) , and h(t) are components functions of r(t) , and

x, y, and z are parametric equations of space curve C.

Write the equation for surface of cone.

z=x2+y2

Substitute 1+y for z,

1+y=x2+y2

Apply square on both sides of equation.

(1+y)2=(x2+y2)21+y2+2y=x2+y22y=x2+y2y21

y=12(x21) (2)

Consider parametric equation x=t .

Substitute t for x in equation (2),

y=12(t21)

Write the equation for surface of plane

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