Hey I need some help with this question



I am having trouble finding the intersection and parametrizing it.

Any insight is much appreciated

Thanks!

x²+y² = 1 is a vertical cylinder,
z = ax + by is a non vertical plane, passing through the origin.

Hence, C is an ellipse.

You can use the following coordinates for a random point of C :

x = cos θ
y = sin θ
z = a.cos θ + b.sin θ

dx = - sin θ.dθ
dy = cos θ.dθ
dz = a.dx + b.dy

Just compute the integral, it's fairly easy.

If I did no mistake, Integral = 2.pi.( a - 1 )

Integral = 0 if and only if a = 1.

How did you come up with this? Is this just a general parametrization for points on an ellipse?

u have x^2+y^2=1

so normally u set x= rcos(theta) and y = rsin(theta)

u also know z=ax+by

sub rsin(theta) and rcos(theta) to get

z = acos(theta) + bsin(theta)

Wow thats great

Thank you both for the help!!

this, therefore b == 0

This post was edited by 2wo1ne on Feb 12 2015 08:11pm

Sorry one more thing but when it says find all values of the two real numbers a,b such that a^2 + b^2 = 1

Is that already given as part of the parametrization? Or do I actually need to find those somehow? Kind of lost at that point

EDIT:

Looking at it again I get

a^2 + b^2 = 1 = x^2 + y^2 so would I say all (x,y) that are on the unit circle satisfy this?


This post was edited by Bloo_Guardian on Feb 12 2015 08:25pm

when u evaluate the integral u get 2pi(a-1).

the question asks for which a,b does this integral = 0
so set 2pi(a-1) = 0
=> a=1
going back to a^2+b^2 =1
1 + b^2 = 1
b = 0

so the only values are a=1 b=0

Ah of course, I was hung up on satsifying a^2 + b^2 = 1 before computing the integral at all

Thank you!