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Question: A point moves so that the sum of its distances from the points (4, 0, 0) and (–4, 0, 0) remains 10. ...

A point moves so that the sum of its distances from the points (4, 0, 0) and (–4, 0, 0) remains 10. The locus of the point is

A

9x2−25y2+25z2=2259x^{2} - 25y^{2} + 25z^{2} = 225

B

9x2+25y2−25z2=2259x^{2} + 25y^{2} - 25z^{2} = 225

C

9x2+25y2+25z2=2259x^{2} + 25y^{2} + 25z^{2} = 225

D

9x2+25y2+25z2+225=09x^{2} + 25y^{2} + 25z^{2} + 225 = 0

Answer

9x2+25y2+25z2=2259x^{2} + 25y^{2} + 25z^{2} = 225

Explanation

Solution

(x−4)2+y2+z2+(x+4)2+y2+z2=10\sqrt { ( x - 4 ) ^ { 2 } + y ^ { 2 } + z ^ { 2 } } + \sqrt { ( x + 4 ) ^ { 2 } + y ^ { 2 } + z ^ { 2 } } = 10

⇒2(x2+y2+z2)+2[(x−4)2+y2+z2][(x+4)2+y2+z2]\Rightarrow 2 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) + 2 \sqrt { \left[ ( x - 4 ) ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right] \left[ ( x + 4 ) ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right] }

=100−32=68= 100 - 32 = 68

⇒(x2+y2+z2−34)2\Rightarrow \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 34 \right) ^ { 2 }

=[(x−4)2+y2+z2][(x+4)2+y2+z2]= \left[ ( x - 4 ) ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right] \left[ ( x + 4 ) ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right]

⇒(x2+y2+z2)2−68(x2+y2+z2)+(34)2\Rightarrow \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } - 68 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) + ( 34 ) ^ { 2 }

=[(x2+y2+z2+16)−8x][(x2+y2+z2+16)+8x]= \left[ \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 16 \right) - 8 x \right] \left[ \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 16 \right) + 8 x \right]

=(x2+y2+z2+16)2−64x2= \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 16 \right) ^ { 2 } - 64 x ^ { 2 }

=(x2+y2+z2)+32(x2+y2+z2)−64x2+(16)2= \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) + 32 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) - 64 x ^ { 2 } + ( 16 ) ^ { 2 }

⇒9x2+25y2+25z2−225=0\Rightarrow 9 x ^ { 2 } + 25 y ^ { 2 } + 25 z ^ { 2 } - 225 = 0 .