Final answer:
The question pertains to finding the equations for a sphere and a plane where a given parametric curve lies, which requires algebraic manipulation to remove the parameter and identify the spatial relationships.
Step-by-step explanation:
The given parametric curve X(T)=2+6cost, Y(T)=4+8cost, Z(T)=2+10sint can be thought of as a spatial curve. To find the sphere's equation, we need to eliminate the parameter 't' and create an equation only in terms of x, y, and z.
For a general sphere, the equation has the form (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, where (a, b, c) is the center of the sphere, and r is its radius. Similarly, a plane has a general form Ax + By + Cz + D = 0, where A, B, and C are the coefficients that define the plane's orientation in space, and D is the distance along the normal from the origin to the plane.
To find the equations of the sphere and plane where the curve lies, we look at the relationships among x, y, and z given by the parametric equations. By combining and manipulating these equations, it might be possible to derive the sphere's and plane's equations that intersect at this curve. However, without doing the appropriate algebra, we cannot give specific forms for the equations of the sphere and plane.