1) I have a question about 2 problem: A circle C of radius 2pi and has a spiral

Question

1) I have a question about 2 problem: A circle C of radius 2pi and has a spiral parametrized by r(t) = (tcost, tsint). Show that the tangent line to the spiral at the point where it intersects C will intersect the y-axis at a distance from the orgin equal to the circumference of C.

I have l(t) = r(2pi) + tr'(2pi)

=(2pi,0) + t(1,2pi)

=(2pi + t, t2pi)

x(t) = 0 -> t= -2pi

c=2pi(2pi)

=4pi^2

I'm not sure if this is correct, if so or if not, can someone explain to me how to do this. If this is correct, can you use your way, I do not really know if my way makes sense.

2) Let f: R^2 -> R^2 be given by f(x,y) = (P(x,y), Q(x,y)) and r(t) = (x(t),y(t)). Find an expression for the tangent vector (f o r)'(t).

It tells me to multiply it out, but I do not have any idea on how to approach this. Any help will be appreciated. Thanks

Explanation / Answer

hi, I was able to find solution to 1st question, if u want you can decrease the points.

1. ) the spiral and the circle will intersect at t=2pi, hence the cordinates of the intersection point is (2pi, 0).

The slope of the tangent is dy/dx = (dy/dt) / (dx/dt) = (sint + tcost) / (cost - tsint) = 2pi , when solved at (2pi, 0)

from slope, we can find the equation of tangent, which is given by

y=2pi*x + c

as tangent passes through (2pi, 0) , it should satisfy the above equation,

therefore, c= -4pi^2

the tangent intersects the y axis at x=0, subsituting x = 0 in the equation,

y = 2pi * (0) - 4pi^2

therefore, y= -4pi^2

which implies, distance of the y intersect from origin = 4pi^2 = circumfrence of circle C.

I believe your method might not be right. You should discuss your method with your professor.

2). (f o r)'(t) = f(r(t))' r(t)' by chain rule, see if you can solve the problem now.

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