Find equations of the tangent plane and normal line to the surface x=6y^2+5z^2−459 at the point (2, 6, 7).
Find equations of the tangent plane and normal line to the surface x=6y^2+5z^2−459 at the point (2, 6, 7).
Equation of the tangent plane (make the coefficient of x equal to 1):
=0.
+t⟨1, , ⟩.
HINT. Represent the given surface as a level surface of a certain function f(x,y,z), that is, re-write the equation of the surface in the form �(�,�,�)=c, where �(�,�,�) is a function of 3 variables and f is a constant. Then find the gradient vector of the function f at the given point. The tangent plane to the surface at that point is orthogonal to the gradient vector, and the normal line -- parallel to it. This allows us to write the equations of both tangent plane and normal line.
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