Consider the paraboloid z = (x^2) + 3y^2 and the plane z = x + y +4 which inters

Question

Consider the paraboloid z = (x^2) + 3y^2 and the plane z = x + y +4 which intersects the paraboloid in a curve C at (2,1,7). Find the equation of the line tangent to C at the point (2,1,7). Proceed as follows.

a. Find a vector normal to the plane at (2,1,7)

b. Find a vector normal to the plane tangent to the paraboloid at (2,1,7)

c. Argue that the line tangent to C at (2,1,7) is orthogonal to both normal vectors found in parts (a) and (b). Use this face to fina a direction vector for the tangent line.

d. Knowing a point on the tangent line and the direction of the tangent line, write an equation of the tangent line in parametric form.

(problem 12.7.52 in Calculus Early Transcendentals by Briggs and Cochran)

Explanation / Answer

https://docs.google.com/viewer?a=v&q=cache:UUnHCbH97rQJ:www.math.rochester.edu/people/faculty/hladky/164s07/oldmidtermsol.pdf+&hl=en&gl=in&pid=bl&srcid=ADGEESjxucYCQQU_RQ4xWtu5IUM2_9ZAdP1dURrpzUq-MlXxb5CBg1p7mPthYHZx4rLGmFr3u3jap84wL2nv9PqH4XZB5fFvnQX587TOScGgbVslOtzK5sp42jGtJRjatSZ1drJHD-HM&sig=AHIEtbQFEwzHQ3pqdMAaUFyFRHhNzYnM1w

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