   Chapter 14.6, Problem 60E

Chapter
Section
Textbook Problem

# At what points does the normal line through the point (1, 2, 1) on the ellipsoid 4x2 + y2 + 4z2 = 12 intersect the sphere x3 + y2 + z3 = 102?

To determine

To find: The ellipsoid 4x2+y2+4z2=12 at (1,2,1) intersects the sphere x2+y2+z2=102 at which point.

Explanation

Result used:

“The normal to the level surface at the point P(x0,y0,z0) is defined as (xx0)Fx(x0,y0,z0)=(yy0)Fy(x0,y0,z0)=(zz0)Fz(x0,y0,z0) ”.

Given:

The equation of the ellipsoid is, 4x2+y2+4z2=12 .

Calculation:

Compute the normal vector to the ellipsoid is, F(x,y,z)=4x2+y2+4z212 .

F(x,y,z)=Fx,Fy,Fz=x(4x2+y2+4z212),y(4x2+y2+4z212),z(4x2+y2+4z212)=(8x),(2y),(8z)

Thus, the normal vector of the paraboloid at the point (1,2,1) is, F(1,2,1)=8,4,8 .

By above result, the normal line is defined as, (x1)Fx(1,2,1)=(y2)Fy(1,2,1)=(z1)Fz(1,2,1) .

The value of Fx(1,2,1)=8 , Fy(1,2,1)=4 and Fz(1,2,1)=8 .

Thus, (x1)8=(y2)4=(z1)8 .

Let, (x1)8=(y2)4=(z1)8=t

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