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Consider the spheres x2 + y2 + z2 - 4y + 3 = 0 and x2 + y2 + z2 + 2x + 4 z - 4 = 0.
What is the distance between the centres of the two spheres ?
- a)5 units
- b)4 units
- c)3 units
- d)2 units
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Consider the spheres x2 + y2 + z2 - 4y + 3 = 0 and x2 + y2 + z2 + 2x +...
x2 + y2 + z2- 4y + 3 = 0
x2 + y2 - 4y+4 — 4 + z2+ 3 = 0
x2 + ( y - 2 )2 + z2 = 1 ...
(i) Sphere with centre (0,2,0) and radius 1 unit.
x2 + y2 + z2 + 2x + 4z - 4 = 0
x2 + 2 x + l-l+ y2 + z2 + 4z + 4 - 4 -4 = 0
(x + l)2 + y2 + (z + 2)2 = 32 ...(ii)
Sphere with centre (-1,0, -2) and radius 3 units.
x2 + y2 - 4y+4 — 4 + z2+ 3 = 0
x2 + ( y - 2 )2 + z2 = 1 ...
(i) Sphere with centre (0,2,0) and radius 1 unit.
x2 + y2 + z2 + 2x + 4z - 4 = 0
x2 + 2 x + l-l+ y2 + z2 + 4z + 4 - 4 -4 = 0
(x + l)2 + y2 + (z + 2)2 = 32 ...(ii)
Sphere with centre (-1,0, -2) and radius 3 units.
Most Upvoted Answer
Consider the spheres x2 + y2 + z2 - 4y + 3 = 0 and x2 + y2 + z2 + 2x +...
To find the distance between the centers of the two spheres, we need to first find the coordinates of the centers of the spheres.
Given Spheres:
1) x^2 + y^2 + z^2 - 4y + 3 = 0 ...(1)
2) x^2 + y^2 + z^2 + 2x + 4z - 4 = 0 ...(2)
To find the center of Sphere 1, we can write the equation of the sphere in the form (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, where (a,b,c) represents the coordinates of the center and r represents the radius.
1) x^2 + (y^2 - 4y) + z^2 + 3 = 0
x^2 + (y^2 - 4y + 4) + z^2 + 3 = 4
x^2 + (y - 2)^2 + z^2 = 1
Comparing this equation with the standard form, we can see that the center of Sphere 1 is at (0, 2, 0) and the radius is 1.
To find the center of Sphere 2, we can write the equation of the sphere in the form (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2.
2) x^2 + (y^2 + 2x) + z^2 + 4z = 4
(x^2 + 2x + 1) + (y^2) + (z^2 + 4z + 4) = 9
(x + 1)^2 + y^2 + (z + 2)^2 = 9
Comparing this equation with the standard form, we can see that the center of Sphere 2 is at (-1, 0, -2) and the radius is 3.
Therefore, the coordinates of the centers of the two spheres are (0, 2, 0) and (-1, 0, -2).
The distance between two points (x1, y1, z1) and (x2, y2, z2) can be calculated using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Calculating the distance between the centers of the two spheres:
d = sqrt((0 - (-1))^2 + (2 - 0)^2 + (0 - (-2))^2)
= sqrt(1 + 4 + 4)
= sqrt(9)
= 3
Therefore, the distance between the centers of the two spheres is 3 units. Hence, option C is the correct answer.
Given Spheres:
1) x^2 + y^2 + z^2 - 4y + 3 = 0 ...(1)
2) x^2 + y^2 + z^2 + 2x + 4z - 4 = 0 ...(2)
To find the center of Sphere 1, we can write the equation of the sphere in the form (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, where (a,b,c) represents the coordinates of the center and r represents the radius.
1) x^2 + (y^2 - 4y) + z^2 + 3 = 0
x^2 + (y^2 - 4y + 4) + z^2 + 3 = 4
x^2 + (y - 2)^2 + z^2 = 1
Comparing this equation with the standard form, we can see that the center of Sphere 1 is at (0, 2, 0) and the radius is 1.
To find the center of Sphere 2, we can write the equation of the sphere in the form (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2.
2) x^2 + (y^2 + 2x) + z^2 + 4z = 4
(x^2 + 2x + 1) + (y^2) + (z^2 + 4z + 4) = 9
(x + 1)^2 + y^2 + (z + 2)^2 = 9
Comparing this equation with the standard form, we can see that the center of Sphere 2 is at (-1, 0, -2) and the radius is 3.
Therefore, the coordinates of the centers of the two spheres are (0, 2, 0) and (-1, 0, -2).
The distance between two points (x1, y1, z1) and (x2, y2, z2) can be calculated using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Calculating the distance between the centers of the two spheres:
d = sqrt((0 - (-1))^2 + (2 - 0)^2 + (0 - (-2))^2)
= sqrt(1 + 4 + 4)
= sqrt(9)
= 3
Therefore, the distance between the centers of the two spheres is 3 units. Hence, option C is the correct answer.
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Consider the spheres x2 + y2 + z2 - 4y + 3 = 0 and x2 + y2 + z2 + 2x + 4 z - 4 = 0.What is the distance between the centres of the two spheres ?a)5 unitsb)4 unitsc)3 unitsd)2 unitsCorrect answer is option 'C'. Can you explain this answer?
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Consider the spheres x2 + y2 + z2 - 4y + 3 = 0 and x2 + y2 + z2 + 2x + 4 z - 4 = 0.What is the distance between the centres of the two spheres ?a)5 unitsb)4 unitsc)3 unitsd)2 unitsCorrect answer is option 'C'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider the spheres x2 + y2 + z2 - 4y + 3 = 0 and x2 + y2 + z2 + 2x + 4 z - 4 = 0.What is the distance between the centres of the two spheres ?a)5 unitsb)4 unitsc)3 unitsd)2 unitsCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the spheres x2 + y2 + z2 - 4y + 3 = 0 and x2 + y2 + z2 + 2x + 4 z - 4 = 0.What is the distance between the centres of the two spheres ?a)5 unitsb)4 unitsc)3 unitsd)2 unitsCorrect answer is option 'C'. Can you explain this answer?.
Consider the spheres x2 + y2 + z2 - 4y + 3 = 0 and x2 + y2 + z2 + 2x + 4 z - 4 = 0.What is the distance between the centres of the two spheres ?a)5 unitsb)4 unitsc)3 unitsd)2 unitsCorrect answer is option 'C'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider the spheres x2 + y2 + z2 - 4y + 3 = 0 and x2 + y2 + z2 + 2x + 4 z - 4 = 0.What is the distance between the centres of the two spheres ?a)5 unitsb)4 unitsc)3 unitsd)2 unitsCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the spheres x2 + y2 + z2 - 4y + 3 = 0 and x2 + y2 + z2 + 2x + 4 z - 4 = 0.What is the distance between the centres of the two spheres ?a)5 unitsb)4 unitsc)3 unitsd)2 unitsCorrect answer is option 'C'. Can you explain this answer?.
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