   Chapter 14.3, Problem 99E

Chapter
Section
Textbook Problem

The ellipsoid 4x2 + 2y2 + z2 = 16 intersects the plane y = 2 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1, 2, 2).

To determine

To find: The parametric equations for the tangent line to the ellipse at the point (1,2,2) .

Explanation

Given:

The ellipsoid 4x2+2y2+z2=16 intersects the plane y=2 in an ellipse.

Calculation:

The equation of the ellipse is 4x2+2y2+z2=16 .

Substitute y=2 in the ellipse equation,

4x2+2(2)2+z2=164x2+4y2+z2=16

By implicit differentiation method differentiate the above equation,

x(4x2+4y2+z2)=x(16)x(4x2)+x(4y2)+x(z2)=04(2x)+0+2zzx=08x+2zzx=0

Simplify further as follows

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