   Chapter 13.1, Problem 31E

Chapter
Section
Textbook Problem

# At what points does the curve r(t) = t i + (2t − t2) k intersect the paraboloid z = x2 + y2?

To determine

The points of intersection of curve r(t)=ti+(2tt2)k and paraboloid z=x2+y2 .

Explanation

Given data:

The vector equation r(t)=ti+(2tt2)k and equation of paraboloid z=x2+y2 .

Definition:

Consider a vector function as r(t)=f(t)i+g(t)j+h(t)k , then parametric equations to plot space curve C are,

x=f(t)y=g(t)z=h(t)

Here,

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

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

From definition, write the parametric equations for vector function r(t)=ti+(2tt2)k .

x=ty=0z=2tt2

Write the equation of paraboloid.

z=x2+y2

Substitute t for x, 0 for y, and 2tt2 for z,

2tt2=t2+(0)2t2+t22t=02t22t=0t(2t2)=0

Hence, there are two possibilities for t value. One is t=0 and the other is,

2t2=02t=2t=22t=1

Therefore, values of t are 0 and 1

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