   SEARCH HOME Math Central Quandaries & 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. Pardeep,

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

Harley

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.?

Pardeep,

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.

Harley     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.