Gradient Vector, Tangent Planes and Normal Lines — Question 5

Question 5 – Solution

Strategy. Parameterize the normal line using the gradient, then substitute that line into the sphere equation.

Step 1: Normal line Let $F=xy+yz+zx$. Then $$\nabla F=\left\langle y+z,x+z,x+y\right\rangle, \qquad \nabla F(P)=\left\langle 2,2,2\right\rangle.$$ Using the simplified direction $\left\langle 1,1,1\right\rangle$, $$\boxed{(x,y,z)=(1+t,1+t,1+t)}.$$

Step 2: Intersections with the sphere Substitution gives $$3(1+t)^2=12,$$ so $(1+t)^2=4$ and $$t=1\quad\text{or}\quad t=-3.$$ The intersection points are therefore $$\boxed{(2,2,2)\quad\text{and}\quad(-2,-2,-2)}.$$

Step 3: Verification The base point satisfies $1+1+1=3$, and the line direction is parallel to $\nabla F(P)$, so the line is normal to the original surface. Finally, $$2^2+2^2+2^2=12, \qquad (-2)^2+(-2)^2+(-2)^2=12,$$ confirming both sphere intersections.