Question

The locus of a point P which moves such that $P{A}^2-P{B}^2=2k^2$ where A and B are $(3,4,5)$ and $(-1,3,-7)$ respectively is

• A. $8x+2y+24z-9+2k^2=0$
• B. $8x+2y+24z-2k^2=0$
• C. $8x+2y+24z+9+2k^2=0$
• D. $8x-2y+24z-2k^2=0$

Answer 1

The correct option is C $8x+2y+24z+9+2k^2=0$

$P{A}^2-P{B}^2=2k^2$

$\Rightarrow [{(x-3)}^2+{(y-4)}^2+{(z-5)}^2]-[{(x+1)}^2+{(y-3)}^2+{(z+7)}^2]=2k^2$

$\Rightarrow (x-3+x+1)(x-3-x-1)+(y-4+y-3)(y-4-y+3)+(z-5+z+7)(z-5-z-7)=2k^2$

$\Rightarrow (2x-2)(-4)+(2y-7)(-1)+(2z+1)(-12)=2k^2$

$\Rightarrow -8x+8-2y+7-24z-24=2k^2$

$\Rightarrow -8x-2y-24z-9=2k^2$

$\Rightarrow 8x+2y+24z+9+2k^2=0$