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Find the equation of the set of point P ,the sum of whose distances from A(4,0,0) B(-4,0,0) is equal to10 ?
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Find the equation of the set of point P ,the sum of whose distances fr...
Let the coordinates of P be (x, y, z).
The coordinates of points A and B are (4, 0, 0) and (–4, 0, 0) respectively.
It is given that PA + PB = 10.
On squaring both sides again, we obtain
25 (x2 + 8x + 16 + y2 + z2) = 625 + 16x2 + 200x
⇒ 25x2 + 200x + 400 + 25y2 + 25z2 = 625 + 16x2 + 200x
⇒ 9x2 + 25y2 + 25z2 – 225 = 0
Thus, the required equation is 9x2 + 25y2 + 25z2 – 225 = 0.
This question is part of UPSC exam. View all Commerce courses
Most Upvoted Answer
Find the equation of the set of point P ,the sum of whose distances fr...
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
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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Find the equation of the set of point P ,the sum of whose distances from A(4,0,0) B(-4,0,0) is equal to10 ?
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Find the equation of the set of point P ,the sum of whose distances from A(4,0,0) B(-4,0,0) is equal to10 ? for Commerce 2024 is part of Commerce preparation. The Question and answers have been prepared according to the Commerce exam syllabus. Information about Find the equation of the set of point P ,the sum of whose distances from A(4,0,0) B(-4,0,0) is equal to10 ? covers all topics & solutions for Commerce 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the equation of the set of point P ,the sum of whose distances from A(4,0,0) B(-4,0,0) is equal to10 ?.
Find the equation of the set of point P ,the sum of whose distances from A(4,0,0) B(-4,0,0) is equal to10 ? for Commerce 2024 is part of Commerce preparation. The Question and answers have been prepared according to the Commerce exam syllabus. Information about Find the equation of the set of point P ,the sum of whose distances from A(4,0,0) B(-4,0,0) is equal to10 ? covers all topics & solutions for Commerce 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the equation of the set of point P ,the sum of whose distances from A(4,0,0) B(-4,0,0) is equal to10 ?.
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