Final answer:
The equation forms either a circle on the xy-plane when z=0 or an infinite cylinder along the z-axis without restriction on z. The standard form is (x - 3)² + (y - 5)² + z = 9 for the cylinder or (x - 3)² + (y - 5)² = 9 for the circle.
Step-by-step explanation:
To express the equation x² + y² − 6x − 10y + z + 25 = 0 in standard form, we need to complete the square for both x and y terms. We do this by adding and subtracting the same values to keep the equation balanced. First, we group the x terms and y terms:
(x² - 6x + 9) + (y² - 10y + 25) - 9 - 25 + z + 25 = 0
The numbers 9 and 25 are chosen because they are squares of half the coefficients of x and y respectively, thus:
(x - 3)² + (y - 5)² + z = 9 (subtract 9 and 25 from both sides)
Now if we are looking at this equation as a geometrical figure, the presence of x² and y² suggests that it can be a circle or some form of a conic section but since z is here as well and is not squared, it can modify the figure into a cylinder if there are no constraints on z. If we set z = 0, it becomes:
(x - 3)² + (y - 5)² = 9
This is the equation of a circle centered at (3,5) with radius 3 on the xy-plane. If z can be any real number, this describes a cylinder with the same circle as its base extending infinitely along the z-axis.
To trace and graph it would depend on whether we restrict z. If z is held constant, it's a circle on the xy-plane. If z is free, we would depict it as a cylinder extended along the z-axis in a three-dimensional space.