Math CentralQuandaries & Queries


Question from pardeep, a student:

we have to show that the curve r(t)=(cos t)i+(sin t)j+(1-cos t)k ,0<=t<=2pie;
is an ellipse by showing it to an intersection of a right circular cylinder and a plane.
i got the eqn. of the cylinder but did not get the eqn of plane.


What is the relationship between the x-coordinate and the z-coordinate of a point on the curve?


Pardeep wrote back

i got the relationship between the x and z coordinate from r(t) which is x+z=1 and it stands to be the plane's eqn..
but i can't understand why we are looking relationship between x and z coordinates on the curve to get a plane's eqn.?
please help.


What you showed is that every point $(x, y, z)$ on the curve satisfies $x + z = 1.$ Hence every point on the curve lies in the plane $x + z = 1.$ Thus every point on the curve lies on the cylinder you found earlier and also on the plane $x + z = 1$ and hence the curve lies on the intersection of the cylinder and the plane.


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