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Equations of spheres practice problems

Spheres: the standard equation from a center and radius, and the center and radius from a general equation. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

20 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 9.611

easy
Find the center and radius of the sphere \( \displaystyle x^{2} + 2 x + y^{2} + 8 y + z^{2} + 8 z + 17 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} + 2 x + y^{2} + 8 y + z^{2} + 8 z + 17 = \left(x + 1\right)^{2} + \left(y + 4\right)^{2} + \left(z + 4\right)^{2} - 16 \]
    So (x − -1)² + (y − -4)² + (z − -4)² = 16.✓ Proved
Answer \( \text{center } (-1, -4, -4),\ \text{radius } 4 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.613

easy
Find the center and radius of the sphere \( \displaystyle x^{2} - 4 x + y^{2} + z^{2} + 8 z + 19 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} - 4 x + y^{2} + z^{2} + 8 z + 19 = y^{2} + \left(x - 2\right)^{2} + \left(z + 4\right)^{2} - 1 \]
    So (x − 2)² + (y − 0)² + (z − -4)² = 1.✓ Proved
Answer \( \text{center } (2, 0, -4),\ \text{radius } 1 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.614

easy
Find the center and radius of the sphere \( \displaystyle x^{2} + 4 x + y^{2} - 4 y + z^{2} - 6 z + 13 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} + 4 x + y^{2} - 4 y + z^{2} - 6 z + 13 = \left(x + 2\right)^{2} + \left(y - 2\right)^{2} + \left(z - 3\right)^{2} - 4 \]
    So (x − -2)² + (y − 2)² + (z − 3)² = 4.✓ Proved
Answer \( \text{center } (-2, 2, 3),\ \text{radius } 2 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.617

easy
Find the center and radius of the sphere \( \displaystyle x^{2} + y^{2} + 10 y + z^{2} - 10 z + 25 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} + y^{2} + 10 y + z^{2} - 10 z + 25 = x^{2} + \left(y + 5\right)^{2} + \left(z - 5\right)^{2} - 25 \]
    So (x − 0)² + (y − -5)² + (z − 5)² = 25.✓ Proved
Answer \( \text{center } (0, -5, 5),\ \text{radius } 5 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.619

easy
Find the center and radius of the sphere \( \displaystyle x^{2} + 8 x + y^{2} + 2 y + z^{2} + 8 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} + 8 x + y^{2} + 2 y + z^{2} + 8 = z^{2} + \left(x + 4\right)^{2} + \left(y + 1\right)^{2} - 9 \]
    So (x − -4)² + (y − -1)² + (z − 0)² = 9.✓ Proved
Answer \( \text{center } (-4, -1, 0),\ \text{radius } 3 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.620

easy
Find the center and radius of the sphere \( \displaystyle x^{2} + 4 x + y^{2} + 8 y + z^{2} - 2 z + 20 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} + 4 x + y^{2} + 8 y + z^{2} - 2 z + 20 = \left(x + 2\right)^{2} + \left(y + 4\right)^{2} + \left(z - 1\right)^{2} - 1 \]
    So (x − -2)² + (y − -4)² + (z − 1)² = 1.✓ Proved
Answer \( \text{center } (-2, -4, 1),\ \text{radius } 1 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.621

easy
Find the center and radius of the sphere \( \displaystyle x^{2} + 4 x + y^{2} - 2 y + z^{2} + 6 z + 5 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} + 4 x + y^{2} - 2 y + z^{2} + 6 z + 5 = \left(x + 2\right)^{2} + \left(y - 1\right)^{2} + \left(z + 3\right)^{2} - 9 \]
    So (x − -2)² + (y − 1)² + (z − -3)² = 9.✓ Proved
Answer \( \text{center } (-2, 1, -3),\ \text{radius } 3 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.623

easy
Find the center and radius of the sphere \( \displaystyle x^{2} - 4 x + y^{2} + 2 y + z^{2} + 4 z = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} - 4 x + y^{2} + 2 y + z^{2} + 4 z = \left(x - 2\right)^{2} + \left(y + 1\right)^{2} + \left(z + 2\right)^{2} - 9 \]
    So (x − 2)² + (y − -1)² + (z − -2)² = 9.✓ Proved
Answer \( \text{center } (2, -1, -2),\ \text{radius } 3 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.698

easy
Find the equation of the sphere with center \( \displaystyle (2, -3, 3) \) and radius \( \displaystyle 3 \), in standard form.
  1. A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
    Reviewed
  2. \[ \left. \left(x - 2\right)^{2} + \left(y + 3\right)^{2} + \left(z - 3\right)^{2} \right|_{\substack{ x=5\\ y=-3\\ z=3 }} = 9 \]
    A point at distance r along the x-direction is on it.✓ Proved
Answer \( \left(x - 2\right)^{2} + \left(y + 3\right)^{2} + \left(z - 3\right)^{2} = 9 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.699

easy
Find the center and radius of the sphere \( \displaystyle x^{2} - 8 x + y^{2} + 8 y + z^{2} + 2 z + 17 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} - 8 x + y^{2} + 8 y + z^{2} + 2 z + 17 = \left(x - 4\right)^{2} + \left(y + 4\right)^{2} + \left(z + 1\right)^{2} - 16 \]
    So (x − 4)² + (y − -4)² + (z − -1)² = 16.✓ Proved
Answer \( \text{center } (4, -4, -1),\ \text{radius } 4 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.701

easy
Find the equation of the sphere with center \( \displaystyle (-4, -3, -3) \) and radius \( \displaystyle 5 \), in standard form.
  1. A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
    Reviewed
  2. \[ \left. \left(x + 4\right)^{2} + \left(y + 3\right)^{2} + \left(z + 3\right)^{2} \right|_{\substack{ x=1\\ y=-3\\ z=-3 }} = 25 \]
    A point at distance r along the x-direction is on it.✓ Proved
Answer \( \left(x + 4\right)^{2} + \left(y + 3\right)^{2} + \left(z + 3\right)^{2} = 25 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.702

easy
Find the center and radius of the sphere \( \displaystyle x^{2} + y^{2} - 8 y + z^{2} - 2 z + 8 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} + y^{2} - 8 y + z^{2} - 2 z + 8 = x^{2} + \left(y - 4\right)^{2} + \left(z - 1\right)^{2} - 9 \]
    So (x − 0)² + (y − 4)² + (z − 1)² = 9.✓ Proved
Answer \( \text{center } (0, 4, 1),\ \text{radius } 3 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.704

easy
Find the center and radius of the sphere \( \displaystyle x^{2} - 2 x + y^{2} - 10 y + z^{2} - 6 z + 10 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} - 2 x + y^{2} - 10 y + z^{2} - 6 z + 10 = \left(x - 1\right)^{2} + \left(y - 5\right)^{2} + \left(z - 3\right)^{2} - 25 \]
    So (x − 1)² + (y − 5)² + (z − 3)² = 25.✓ Proved
Answer \( \text{center } (1, 5, 3),\ \text{radius } 5 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.707

easy
Find the center and radius of the sphere \( \displaystyle x^{2} - 4 x + y^{2} + 6 y + z^{2} - 2 z - 22 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} - 4 x + y^{2} + 6 y + z^{2} - 2 z - 22 = \left(x - 2\right)^{2} + \left(y + 3\right)^{2} + \left(z - 1\right)^{2} - 36 \]
    So (x − 2)² + (y − -3)² + (z − 1)² = 36.✓ Proved
Answer \( \text{center } (2, -3, 1),\ \text{radius } 6 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.798

easy
Find the equation of the sphere with center \( \displaystyle (2, -2, -5) \) and radius \( \displaystyle 2 \), in standard form.
  1. A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
    Reviewed
  2. \[ \left. \left(x - 2\right)^{2} + \left(y + 2\right)^{2} + \left(z + 5\right)^{2} \right|_{\substack{ x=4\\ y=-2\\ z=-5 }} = 4 \]
    A point at distance r along the x-direction is on it.✓ Proved
Answer \( \left(x - 2\right)^{2} + \left(y + 2\right)^{2} + \left(z + 5\right)^{2} = 4 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.801

easy
Find the center and radius of the sphere \( \displaystyle x^{2} - 10 x + y^{2} + 10 y + z^{2} - 10 z + 59 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} - 10 x + y^{2} + 10 y + z^{2} - 10 z + 59 = \left(x - 5\right)^{2} + \left(y + 5\right)^{2} + \left(z - 5\right)^{2} - 16 \]
    So (x − 5)² + (y − -5)² + (z − 5)² = 16.✓ Proved
Answer \( \text{center } (5, -5, 5),\ \text{radius } 4 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.806

easy
Find the center and radius of the sphere \( \displaystyle x^{2} - 4 x + y^{2} - 2 y + z^{2} + 8 z + 5 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} - 4 x + y^{2} - 2 y + z^{2} + 8 z + 5 = \left(x - 2\right)^{2} + \left(y - 1\right)^{2} + \left(z + 4\right)^{2} - 16 \]
    So (x − 2)² + (y − 1)² + (z − -4)² = 16.✓ Proved
Answer \( \text{center } (2, 1, -4),\ \text{radius } 4 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.839

easy
Find the center and radius of the sphere \( \displaystyle x^{2} - 4 x + y^{2} - 6 y + z^{2} + 12 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} - 4 x + y^{2} - 6 y + z^{2} + 12 = z^{2} + \left(x - 2\right)^{2} + \left(y - 3\right)^{2} - 1 \]
    So (x − 2)² + (y − 3)² + (z − 0)² = 1.✓ Proved
Answer \( \text{center } (2, 3, 0),\ \text{radius } 1 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.843

easy
Find the center and radius of the sphere \( \displaystyle x^{2} + 8 x + y^{2} + 4 y + z^{2} + 6 z - 7 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} + 8 x + y^{2} + 4 y + z^{2} + 6 z - 7 = \left(x + 4\right)^{2} + \left(y + 2\right)^{2} + \left(z + 3\right)^{2} - 36 \]
    So (x − -4)² + (y − -2)² + (z − -3)² = 36.✓ Proved
Answer \( \text{center } (-4, -2, -3),\ \text{radius } 6 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.910

easy
Find the center and radius of the sphere \( \displaystyle x^{2} + 2 x + y^{2} - 4 y + z^{2} - 6 z - 22 = 0 \).
  1. Complete the square in x, y and z.
    Reviewed
  2. \[ x^{2} + 2 x + y^{2} - 4 y + z^{2} - 6 z - 22 = \left(x + 1\right)^{2} + \left(y - 2\right)^{2} + \left(z - 3\right)^{2} - 36 \]
    So (x − -1)² + (y − 2)² + (z − 3)² = 36.✓ Proved
Answer \( \text{center } (-1, 2, 3),\ \text{radius } 6 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More equations of spheres practice

Find the center and radius of the sphere \( \displaystyle x^{2} + y^{2} + z^{2} - 8 z + 15 = 0 \).
Problem 9.612easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} + 4 x + y^{2} + 6 y + z^{2} + 8 z + 28 = 0 \).
Problem 9.615easy✓ Nihil obstat
Find the equation of the sphere with center \( \displaystyle (2, 5, 4) \) and radius \( \displaystyle 5 \), in standard form.
Problem 9.616easy✓ Every equation proved
Find the equation of the sphere with center \( \displaystyle (0, 0, 2) \) and radius \( \displaystyle 3 \), in standard form.
Problem 9.618easy✓ Every equation proved
Find the equation of the sphere with center \( \displaystyle (0, 4, 0) \) and radius \( \displaystyle 5 \), in standard form.
Problem 9.622easy✓ Every equation proved
Find the equation of the sphere with center \( \displaystyle (-2, -1, 2) \) and radius \( \displaystyle 2 \), in standard form.
Problem 9.624easy✓ Every equation proved
Find the equation of the sphere with center \( \displaystyle (3, -4, 1) \) and radius \( \displaystyle 2 \), in standard form.
Problem 9.625easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} + 10 x + y^{2} + 6 y + z^{2} + 8 z + 49 = 0 \).
Problem 9.696easy✓ Nihil obstat
Find the center and radius of the sphere \( \displaystyle x^{2} + 6 x + y^{2} - 2 y + z^{2} + 2 z + 10 = 0 \).
Problem 9.697easy✓ Nihil obstat
Find the equation of the sphere with center \( \displaystyle (-3, -1, -5) \) and radius \( \displaystyle 2 \), in standard form.
Problem 9.700easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} + 6 x + y^{2} + 6 y + z^{2} + 10 z + 18 = 0 \).
Problem 9.703easy✓ Nihil obstat
Find the equation of the sphere with center \( \displaystyle (-1, -1, -1) \) and radius \( \displaystyle 3 \), in standard form.
Problem 9.705easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} + 8 x + y^{2} + 4 y + z^{2} - 8 z + 11 = 0 \).
Problem 9.706easy✓ Nihil obstat
Find the center and radius of the sphere \( \displaystyle x^{2} - 6 x + y^{2} + 4 y + z^{2} - 2 z + 5 = 0 \).
Problem 9.708easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} - 6 x + y^{2} + 4 y + z^{2} + 2 z + 5 = 0 \).
Problem 9.709easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} - 8 x + y^{2} - 8 y + z^{2} - 4 z + 11 = 0 \).
Problem 9.710easy✓ Nihil obstat
Find the equation of the sphere with center \( \displaystyle (-1, 0, 0) \) and radius \( \displaystyle 2 \), in standard form.
Problem 9.797easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} - 6 x + y^{2} + 10 y + z^{2} - 6 z + 34 = 0 \).
Problem 9.799easy✓ Every equation proved
Find the equation of the sphere with center \( \displaystyle (4, 0, 3) \) and radius \( \displaystyle 2 \), in standard form.
Problem 9.800easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} - 10 x + y^{2} - 8 y + z^{2} - 4 z + 44 = 0 \).
Problem 9.802easy✓ Nihil obstat
Find the center and radius of the sphere \( \displaystyle x^{2} + 4 x + y^{2} - 8 y + z^{2} + 4 z + 23 = 0 \).
Problem 9.803easy✓ Nihil obstat
Find the equation of the sphere with center \( \displaystyle (-1, -4, 2) \) and radius \( \displaystyle 1 \), in standard form.
Problem 9.804easy✓ Every equation proved
Find the equation of the sphere with center \( \displaystyle (1, -4, 5) \) and radius \( \displaystyle 3 \), in standard form.
Problem 9.805easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} + y^{2} - 10 y + z^{2} - 10 z + 14 = 0 \).
Problem 9.840easy✓ Nihil obstat
Find the equation of the sphere with center \( \displaystyle (-3, 0, 5) \) and radius \( \displaystyle 4 \), in standard form.
Problem 9.841easy✓ Every equation proved
Find the equation of the sphere with center \( \displaystyle (-5, -2, 0) \) and radius \( \displaystyle 2 \), in standard form.
Problem 9.842easy✓ Every equation proved
Find the equation of the sphere with center \( \displaystyle (5, 1, 3) \) and radius \( \displaystyle 3 \), in standard form.
Problem 9.844easy✓ Every equation proved
Find the equation of the sphere with center \( \displaystyle (4, -5, -1) \) and radius \( \displaystyle 5 \), in standard form.
Problem 9.845easy✓ Every equation proved
Find the equation of the sphere with center \( \displaystyle (2, -2, -5) \) and radius \( \displaystyle 4 \), in standard form.
Problem 9.846easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} + 4 x + y^{2} + 4 y + z^{2} + 8 z + 15 = 0 \).
Problem 9.847easy✓ Nihil obstat
Find the center and radius of the sphere \( \displaystyle x^{2} + y^{2} + 6 y + z^{2} - 10 z + 25 = 0 \).
Problem 9.848easy✓ Nihil obstat
Find the equation of the sphere with center \( \displaystyle (-4, 2, 1) \) and radius \( \displaystyle 1 \), in standard form.
Problem 9.911easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} - 2 x + y^{2} + 10 y + z^{2} - 8 z + 33 = 0 \).
Problem 9.912easy✓ Nihil obstat
Find the equation of the sphere with center \( \displaystyle (0, 3, -4) \) and radius \( \displaystyle 4 \), in standard form.
Problem 9.913easy✓ Every equation proved
Find the equation of the sphere with center \( \displaystyle (1, -3, 5) \) and radius \( \displaystyle 4 \), in standard form.
Problem 9.914easy✓ Every equation proved
Find the center and radius of the sphere \( \displaystyle x^{2} - 4 x + y^{2} - 4 y + z^{2} + 4 = 0 \).
Problem 9.915easy✓ Nihil obstat
Find the center and radius of the sphere \( \displaystyle x^{2} + 4 x + y^{2} + 8 y + z^{2} + 6 z + 20 = 0 \).
Problem 9.916easy✓ Nihil obstat
Find the center and radius of the sphere \( \displaystyle x^{2} + 2 x + y^{2} + 2 y + z^{2} + 6 z + 2 = 0 \).
Problem 9.917easy✓ Nihil obstat
Find the center and radius of the sphere \( \displaystyle x^{2} + 6 x + y^{2} + z^{2} + 6 z + 9 = 0 \).
Problem 9.918easy✓ Nihil obstat
Find the equation of the sphere with center \( \displaystyle (4, 2, 0) \) and radius \( \displaystyle 1 \), in standard form.
Problem 9.919easy✓ Every equation proved