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Chapter 10: Q 37 (page 789)

Use mid point formula to find the equation of the sphere in which the segment with endpoints (6, -1, 4) and (-3, 3, 1) is a diameter.

Short Answer

Expert verified

The required equation is ${(x - \frac{3}{2})}^{2} + {(y - 1)}^{2} + {(z - \frac{5}{2})}^{2} = \frac{53}{2}$

Step by step solution

01

Given information

A sphere in which the segment with endpoints (6, -1, 4) and (-3, 3, 1) is a diameter.

02

Finding the center and radius

The center, as per mid point formula, is:

$(\frac{6 - 3}{2}, \frac{- 1 + 3}{2}, \frac{4 + 1}{2})$

Simplify:

$(\frac{3}{2}, 1, \frac{5}{2})$

The radius will be half of diameter.

The radius is:

$\frac{\sqrt{{(6 + 3)}^{2} + {(- 1 - 3)}^{2} + {(4 - 1)}^{2}}}{2}$

Simplify:

Radius = $\frac{\sqrt{106}}{2}$

03

Finding the equation

The equation of sphere may be written as:

${(x - \frac{3}{2})}^{2} + {(y - 1)}^{2} + {(z - \frac{5}{2})}^{2} = {(\frac{\sqrt{106}}{2})}^{2}$

Simplify:

${(x - \frac{3}{2})}^{2} + {(y - 1)}^{2} + {(z - \frac{5}{2})}^{2} = \frac{53}{2}$

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Most popular questions from this chapter

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The sum formulas in Theorem 4.4 can be applied only to sums whose starting index value is $k = 1$ .

(b) True or False: ${鈭�}_{k = 0}^{n} 鈥� \frac{1}{k + 1} + {鈭�}_{k = 1}^{n} 鈥� k^{2}$ is equal to ${鈭�}_{k = 0}^{n} 鈥� \frac{k^{3} + k^{2} + 1}{k + 1}$ .

(c) True or False: ${鈭�}_{k = 1}^{n} 鈥� \frac{1}{k + 1} + {鈭�}_{k = 0}^{n} 鈥� k^{2}$ is equal to ${鈭�}_{k = 1}^{n} 鈥� \frac{k^{3} + k^{2} + 1}{k + 1}$ .

(d) True or False: $({鈭�}_{k = 1}^{n} 鈥� \frac{1}{k + 1}) ({鈭�}_{k = 1}^{n} 鈥� k^{2})$ is equal to ${鈭�}_{k = 1}^{n} 鈥� \frac{k^{2}}{k + 1}$ .

(e) True or False: ${鈭�}_{k = 0}^{m} 鈥� \sqrt{k} + {鈭�}_{k = m}^{n} 鈥� \sqrt{k}$ is equal to ${鈭�}_{k = 0}^{n} 鈥� \sqrt{k}$ .

(f) True or False: ${鈭�}_{k = 0}^{n} 鈥� a_{k} = 鈭� a_{0} 鈭� a_{n} + {鈭�}_{k = 1}^{n 鈭� 1} 鈥� a_{k}$ .

(g) True or False: ${({鈭�}_{k = 1}^{10} 鈥� a_{k})}^{2} = {鈭�}_{k = 1}^{10} 鈥� a_{k}^{2}$ .

(h) True or False: ${鈭�}_{k = 1}^{n} 鈥� {(e^{x})}^{2} = \frac{e^{x} (e^{x} + 1) (2 e^{x} + 1)}{6}$ .

Find $\overset{鈬赌}{P Q}$

$P = (5, - 3), Q = (3, - 6)$

What is Lagrange鈥檚 identity? How is it used to understand the geometry of the cross product?

Find $u + v$ and $u - v$ . Also sketch $u, v, u + v$ and $u - v$ .

role="math" localid="1649578020551" $u = (3, - 4) a n d v = (- 1, 5)$

In Exercises 22鈥�29 compute the indicated quantities when $u = (2, 1, 鈭� 3), v = (4, 0, 1), and w = (鈭� 2, 6, 5)$

$u 脳 v and v 脳 u$

See all solutions

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