A point `P(x ,y ,z)` is such that `3P A=2P B ,` where `Aa n dB` are the point `(1,3,4)a n d(1,-2,-1),` erespectivley. Find the equation to the locus of the point `P` and verify that the locus is a sphere.

Explanation

Since `3PA= 2PB`, we get `9PA^(2)= 4PB^(2)`
`" " 9[(x-1)^(2)+(y-3)^(2) + (z-4)^(2)]`
`" "= 4[x-1^(2)+ (y+2)^(2)+ (z+1)^(2)]`
`" "9[x^(2)+y^(2)+z^(2)-2x-6y--8z+26]`
`" "=4[c^(2)+y^(2)+z^(2)-2x+4y+2z+6]`
`" "5x^(2)+5y^(2)-10x-70y-80z+210=0`
`" "x^(2)+y^(2)=z^(2)-2x-14y-16z+42=0`
This represents a sphere with centre at (1, 7, 8) and radius equal to `sqrt(1^(2-)+7^(2)+8^(2)-42)= sqrt(72) = 6sqrt(2)`