Question

The equation of the set of point $P$, the sum of whose distances from $A(4,0,0)$ and $B(-4,0,0)$ is equal to $10$, is

• A. $3x^2+25y^2+16z^2-225=0$
• B. $9x^2+25y^2+25z^2-225=0$
• C. $5x^2+16y^2+18z^2-225=0$
• D. None of the above

Answer 1

Answer: (B)

Let the point be $P(x,y,z)$. Then, it is given $PA+PB=10$
$\Rightarrow \sqrt{(x-4{)}^2+(y-0{)}^2+(z-0{)}^2}$
$+\sqrt{(x+4{)}^2+(y-0{)}^2+(z-0{)}^2}=10$
$\Rightarrow \sqrt{(x-4{)}^2+y^2+z^2}=10-\sqrt{(x+4{)}^2+y^2+z^2}$
Squaring on both sides, we get
$(x-4{)}^2+y^2+z^2=100+(x+4{)}^2+y^2+z^2$
$-20\sqrt{(x+4{)}^2+y^2+z^2}$
$\Rightarrow x^2+16-8x=100+x^2+16+8x$
$-20\sqrt{(x+4{)}^2+y^2+z^2}$
$\Rightarrow -8x-8x-100=-20\sqrt{(x+4{)}^2+y^2+z^2}$
$\Rightarrow -16x-100=-20\sqrt{(x+4{)}^2+y^2+z^2}$
$\Rightarrow 4x+25=5\sqrt{(x+4{)}^2+y^2+z^2}$
(dividing both sides by $-4$ )
Again, squaring on both sides, we get
$(4x+25{)}^2=25\left[(x+4{)}^2+y^2+z^2\right]$
$\Rightarrow 16x^2+625+200x=25\left[(x+4{)}^2+y^2+z^2\right]$
$\Rightarrow 16x^2+625+200x=25\left[x^2+16+8x+y^2+z^2\right]$
$\Rightarrow 16x^2+625+200x=25x^2+400+200x+25y^2+25z^2$
$\Rightarrow 9x^2+25y^2+25z^2-225=0$
which is the required equation.