Write as a Vector Equality x^2+y^2=4 , y=x+2

Problem

x2+y2=4,y=x+2

Solution

1. Identify the goal, which is to represent the intersection of the circle x2+y2=4 and the line y=x+2 as a single vector equation r*(t)

2. Parameterize the system by choosing a parameter t Since the line is already solved for y in terms of x let x=t

3. Substitute the parameter into the linear equation to find the component for y

y=t+2

4. Define the vector r*(t) using the components x(t) and y(t)

r*(t)=<t,t+2>

5. Determine the interval for t by substituting the parametric expressions into the circle equation x2+y2=4

t2+(t+2)2=4

6. Solve for the boundary values of t

t2+t2+4*t+4=4

2*t2+4*t=0

2*t*(t+2)=0

7. Conclude that the vector equality represents the segment of the line between the intersection points where t∈[−2,0]

Final Answer

r*(t)=<t,t+2>,−2≤t≤0

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