Final answer:
The largest sphere centered at (5, 3, 6) that fits in the first octant has a radius equal to the smallest coordinate of the center, which is 3. Therefore, the equation of that sphere in a form that's monic is: x^2 - 10x + y^2 - 6y + z^2 - 12z + 59 = 0
Step-by-step explanation:
To find the equation of the largest sphere with center (5, 3, 6), we must first understand what it means to say the sphere is contained in the first octant. The first octant is the part of the 3-dimensional coordinate system where all three coordinates (x, y, z) are positive. The center of our sphere is in the point (5, 3, 6). Given that, the maximum radius this sphere can have without crossing beyond the first octant is equal to the smallest coordinate of the center, which is 3.
Therefore, the equation of the sphere in standard form where the formula is monic and center is at point (5, 3, 6) is: (x - 5)^2 + (y - 3)^2 + (z - 6)^2 = 9. The right side of the equation, 9, is the radius squared.
So the equation of our sphere is: x^2 - 10x + y^2 - 6y + z^2 - 12z + 59 = 0
Learn more about equation of the largest sphere