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Line tangent to an intersection curve Consider the
Chapter 10, Problem 52E(choose chapter or problem)
Line tangent to an intersection curve Consider the paraboloid \(z=x^{2}+3 y^{2}\) and the plane z = x + y + 4, which intersects the paraboloid in a curve Cat (2, 1, 7) (see figure). 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 Cat (2, 1, 7) is orthogonal to both normal vectors found in parts (a) and (b). Use this fact to find 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.
Questions & Answers
QUESTION:
Line tangent to an intersection curve Consider the paraboloid \(z=x^{2}+3 y^{2}\) and the plane z = x + y + 4, which intersects the paraboloid in a curve Cat (2, 1, 7) (see figure). 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 Cat (2, 1, 7) is orthogonal to both normal vectors found in parts (a) and (b). Use this fact to find 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.
ANSWER:
Problem 52EStep 1 of 7Consider the paraboloid equation as shown below: Consider the plane: The above equation intersects at the point C .