Question

At what points does the normal line through the point (1,2,1) on the ellipsoid 4x²+y²+4z²=12 intersect the sphere x²+y²+z²=486 ?

smaller x-value (x, y, z) = _________
larger x-value (x, y, z) = _________

Answer 1

To find the intersection points of the normal line to the ellipsoid at (1, 2, 1) with the given sphere, we calculate the gradient of the ellipsoid at that point, derive the parametric equations of the normal line, and then find where this line intersects the sphere.

Step-by-step explanation:

To determine at which points a normal line to the ellipsoid 4x²+y²+4z²=12 at the point (1, 2, 1) intersects the sphere x²+y²+z²=486, we must first find the gradient of the ellipsoid at the given point, which will give us the direction vector of the normal line. We first calculate the partial derivatives at the point (1, 2, 1).

The gradient of the ellipsoid 4x²+y²+4z²=12 is ∇f = (8x, 2y, 8z). At the point (1, 2, 1), this becomes ∇f(1, 2, 1) = (8, 4, 8). Thus, the normal line can be expressed parametrically as:

• x = 1 + 8t
• y = 2 + 4t
• z = 1 + 8t

To find the intersection points with the sphere x²+y²+z²=486, substitute these parametric equations into the equation of the sphere and solve for t. The resulting quadratic equation in t will yield two values, from which we can find the two intersection points with different x-values. We will sort these points based on their x-values to provide the smaller and larger x-values as required.