How do I find the parametrization of the intersection of a cylinder and a plane?

Let S be the cylinder of radius 5 centered about the z-axis and let P be the plane described by the equation 6x+8y-25z=0. Suppose C is the curve obtained by intersecting S and P. Find a parametrization of the curve C. How do I find the parametrization of the intersection of a cylinder and a plane?You know that in this case you have a cylinder with x^2+y^2=5^2. In the other hand you have plane. In most cases this plane is slanted and so your curve created by the intersection by these two planes will be an ellipse.



The projection of C onto the x-y plane is the circle x^2+y^2=5^2, z=0, so we know that



x=5cos(t) and y=5sin(t)



so from the equation 6x+8y-25z=0 you have z=(6x+8y)/25 but yet you know your x and y. so it becomes



z=[(30cos(t)+40sin(t)]/25



so the parametrization of the curve C is



r(t)= 5cos(t)i+5sin(t)j+[(30cos(t)+40sin(t)]/2鈥?