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Chapter 10: Q 36 (page 789)

Use the midpoint formula to find the equations of the sphere in which the segment with endpoints (3, −2, 6) and (5, 6, 4) is a diameter.

Short Answer

Expert verified

The required equation is ${(x - 4)}^{2} + {(y - 2)}^{2} + {(z - 5)}^{2} = 18$

Step by step solution

01

Given information

A sphere in which the segment with endpoints (3, −2, 6) and (5, 6, 4) is a diameter.

02

Finding the radius and center

The radius is half of diameter which is half of the distance between the two endpoints given.

Radius is:

$\frac{1}{2} \sqrt{{(5 - 3)}^{2} + {(6 + 2)}^{2} + {(4 - 6)}^{2}}$

Solving:

Radius is $3 \sqrt{2}$

The center, as per mid point formula, is:

$(\frac{5 + 3}{2}, \frac{6 - 2}{2}, \frac{4 + 6}{2})$

Simplify:

$(4, 2, 5)$

03

Writing the equation

The equation of sphere is:

${(x - 4)}^{2} + {(y - 2)}^{2} + {(z - 5)}^{2} = {(3 \sqrt{2})}^{2}$

Simplify:

${(x - 4)}^{2} + {(y - 2)}^{2} + {(z - 5)}^{2} = 18$

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Most popular questions from this chapter

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

the formal $δ - M, N - \in,$ , and N–M definitions of the limit statements $\lim_{x \to c} f (x) = \infty, \lim_{x \to \infty} f (x) = L,$ and $\lim_{x \to \infty} f (x) = \infty$ , respectively

In Exercises 36–41 use the given sets of points to find:

(a) A nonzero vector N perpendicular to the plane determined by the points.

(b) Two unit vectors perpendicular to the plane determined by the points.

(c) The area of the triangle determined by the points.

$P (3, 1, 8), Q (0, 6, - 1), R (- 3, 5, - 3)$

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

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

Find $\vec{P Q}$

$P = (- 1, 5, 4), Q = (- 1, 6, 4)$

If u, v and w are three vectors in $ℝ^{3}$ , what is wrong with the expression $u \times v \times w$ ?

See all solutions

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