Find all points on the ellipsoid x2y22z21 where the tangent

Find all points on the ellipsoid x²+y²+2z²=1 where the tangent plane is perpendicular to the line given by < x, y, z > = < t-1, 2t+1, 2-t >
Include sketch/diagram of region of integration.
Start with a \"doodle\" sketch of the ellipse, the line (which does not necessarily touch the ellipsoid), and a tangent plane to the sphere which is perpendicular to the line. Parallel vectors are scalar multiples of one another, and you WILL USE this property to solve the problem correctly.

Solution

Let F(x,y,z) define a surface that is differentiable at a point (x₀,y₀,z₀), then the normal line to F(x,y,z) at ( x₀ , y₀, z₀ ) is the line with normal vector

          GradF(x₀,y₀,z₀)

that passes through the point (x₀,y₀,z₀). In Particular the equation of the normal line is

          x(t) = x₀ + F_x(x₀,y₀,z₀) t  

          y(t) = y₀ + F_y(x₀,y₀,z₀) t  

          z(t) = z₀ + F_z(x₀,y₀,z₀) t

We want (x, y, z) such that x² + 2y² + 2z² = 1 and < x, y, z >= <t-1 ,2t+1, 2-t >, for some , that is x = 3/t-1, y = /2t+1, z = /2-t.

Thus we must have x² + y² + 2z² = (9/(t-1)²+1/(2t+1)²+1/(2-t)²)² =1

On solving the obtained equation we will get the point.