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If A and B be the points (3,4,5) and 9-1...

If A and B be the points `(3,4,5) and 9-1,3,-7)` respectively find the equation of set of ponts P such that `PA^2+PB^2=k^2, where k` is a constant.

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Let the coordinates of moving point P be (x,y,z).
Given, `A-=(3,4,5),B(-1,3,-7)`
According to the problem, `PA^(2)+PB^(2)=k^(2)`
`implies (x-3)^(2)+(y-4)^(2)+(z-5)^(2)+(x+1)^(2)+(y-3)^(2)+(z+7)^(2)=k^(2)`
`implies x^(2)-5x+9+y^(2)+8y+16+z^(2)-10z+25+x^(2)+2x+1+y^(2)-6y+9+z^(2)+14z+49=k^(2)`
`implies 2x^(2)+2y^(2)+2z^(2)-4x-14y+4z=k^(2)-109`
`implies x^(2)+y^(2)+^(2)-2x-7y+2z=(k^(2)-109)/(2)`
Which is the required equation.
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