One end-point of a diameter of the sphere x2 + y2 + z2 - x - 2z = 1 is (1,1,0). Then the other end-
point of the diameter will be
A) (0, 1, 0) B) (1, 1, 2). C) (1,72,1) D) (0,-1,2)
Answers
Given : One end-point of a diameter of the sphere x² + y² + z² - x - 2z = 1 is (1,1,0).
To Find : the other end- point of the diameter
A) (0, 1, 0) B) (1, 1, 2). C) (1,72,1) D) (0,-1,2)
Solution:
x² + y² + z² - x - 2z = 1
=> x² - x + y² + z² - 2z = 1
=> x² - x + 1/4 - 1/4 + y² + z² - 2z + 1 - 1 = 1
=> ( x - 1/2)² + y² + ( z - 1)² = 9/4
=> ( x - 1/2)² + y² + ( z - 1)² = (3/2)²
center of sphere = 1/2 , 0 , 1
One End point of diameter = ( 1 , 1, 0)
Another End point of diameter = ( x , y , z)
Mid point will be center
=> (x + 1)/2 = 1/2 , ( y + 1)/2 = 0 , ( z + 0)/2 = 1
=> x = 0 , y = - 1 , z = 2
( 0 , - 1 , 2) will be other end- point of the diameter
Option D is correct
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Answer:
x² + y² + z² - x - 2z = 1
=> x² - x + y² + z² - 2z = 1
=> x² - x + 1/4 - 1/4 + y² + z² - 2z + 1 - 1 = 1
=> ( x - 1/2)² + y² + ( z - 1)² = 9/4
=> ( x - 1/2)² + y² + ( z - 1)² = (3/2)²
center of sphere = 1/2 , 0 , 1
One end point of the diameter = ( 1 , 1, 0)
Another end point of the diameter = ( x , y , z)
Mid point will be center
=> (x + 1)/2 = 1/2 , ( y + 1)/2 = 0 , ( z + 0)/2 = 1
=> x = 0 , y = - 1 , z = 2
( 0 , - 1 , 2) will be other end- point of the diameter
Option D is correct