Question

Find the equation of the largest sphere S through the point (0,1,3) such that every point on S satisfies

x² + y² + z² < 20 + 2(2x + z)

Answer 1

The largest sphere that passes through the point (0,1,3) and satisfies the given inequality is one centered at (2,0,1) with a radius slightly less than 5, with the equation (x - 2)² + y² + (z - 1)² ≤ 5² being the limiting case.

Step-by-step explanation:

The question asks for the equation of the largest sphere S that passes through the point (0,1,3) and satisfies the inequality x² + y² + z² < 20 + 2(2x + z). To find this sphere's equation, we need to express the inequality as the standard equation of a sphere, which has the form (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) is the center of the sphere and r is its radius.

Rearranging the inequality, we have:

²+ y² + z² - 4x - 2z < 20

Or:

(x - 2)² + y² + (z - 1)² < 25

This suggests a sphere with center (2,0,1) and radius 5. However, since the inequality is strict (<), the sphere's points satisfy x² + y² + z² < 25, indicating that the largest possible radius that still includes the point (0,1,3) and satisfies the inequality is slightly less than 5. Thus, the equation for the largest sphere is:

(x - 2)² + y² + (z - 1)² ≤ 5²

Note: The equation provided is the limiting case; technically, the actual largest radius would be infinitesimally less than 5, but for practical purposes in a high school setting, the radius is taken as 5.