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Chapter 9: Problem 34

Complete the square to write the equation of the sphere in standard form. Find the center and radius. \(x^{2}+y^{2}+z^{2}+9 x-2 y+10 z+19=0\)

Short Answer

Expert verified
The center of the sphere is (-4.5, 1, -5) and the radius of the sphere is \(\sqrt{14.25}\).

Step by step solution

01

Grouping the Terms

Rearrange the given equation to have terms of the same variable grouped together. We rewrite it as \((x^{2}+9x) + (y^{2} - 2y) + (z^{2} + 10 z) + 19 = 0\)
02

Completing the Square and Rewriting Equation

Complete the squares for each group of variable terms. This is done by adding and subtracting the square of half the coefficients of 'x', 'y' and 'z' terms in the equation. The equation now becomes \((x+4.5)^2 - 4.5^2 + (y-1)^2 - 1^2 + (z+5)^2 - 5^2 + 19 = 0\). Simplifying this gives us \((x+4.5)^2 + (y-1)^2 + (z+5)^2 = 4.5^2 + 1^2 + 5^2 - 19\).
03

Calculating Radius and Writing Equation

Simplify the right side of the equation to calculate the radius. This gives us \((x+4.5)^2 + (y-1)^2 + (z+5)^2 = 14.25\). So, the square of the radius is 14.25 which implies radius, r = \(\sqrt{14.25}\).
04

Identifying the Center and Radius

Compare the simplified equation with the standard form to identify the center (h, k, j) and radius r. The center of the sphere is (-4.5, 1, -5) and the radius of the sphere is \(\sqrt{14.25}\).

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