Question:
find the equation of the set of point P, the sum of whose distance from A(4.0.0) and B(-4,0,0) is equal to 10
Answer:
Let the point P is (x,y,z)
Now given that
PA + PB = 10
=> √{(x-4)2 + y2 + z2 } + √{(x+4)2 + y2 + z2 } = 10
=> √{(x-4)2 + y2 + z2 } = 10 - √{(x+4)2 + y2 + z2 }
Now square bothe side
[√{(x-4)2 + y2 + z2 }]2 = (10)2 + [{(x+4)2 + y2 + z2 }]2 - 2 *10*√{(x+4)2 + y2 + z2 }
=> {(x-4)2 + y2 + z2 } = 100 + {(x+4)2 + y2 + z2 } - 20*√{(x+4)2 + y2 + z2 }
=> x2 + 16 - 8x + y2 + z2 = 100 + x2 + 16 + 8x + y2 + z2 - 20*√{(x+4)2 + y2 + z2 }
=> - 8x = 100 + 8x - 20*√{(x+4)2 + y2 + z2 }
=> -8x -8x - 100 = - 20*√{(x+4)2 + y2 + z2 }
=> -16x -100 = - 20*√{(x+4)2 + y2 + z2 }
=> 4x + 25 = 5*√{(x+4)2 + y2 + z2 }
Again square bothe side,
(4x + 25)2 = 25 *[√{(x+4)2 + y2 + z2 }]2
=> 16x2 + 625 + 200x = 25*{(x+4)2 + y2 + z2 }
=> 16x2 + 625 + 200x = 25*(x2 + 16 + 8x + y2 + z2 )
=> 16x2 + 625 + 200x = 25x2 + 400 + 200x + 25y2 + 25z2
=> 25x2 + 400 + 200x + 25y2 + 25z2 - 16x2 - 625 - 200x = 0
=> 9x2 + 25y2 + 25z2 - 225 = 0
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