A point moves so that the ratio of its distances from two fixed points is constant. Its locus is a:

sphere

Let the coordinates of moving point P be (x, y, z). Let A (a, 0, 0) and B (–a, 0, 0) be two fixed points. According to given condition

$\frac{A P}{B P}$ = constant  = k (say)  $\Rightarrow A P^2=k^2 B P^2$

or, $(x-a)^2+(y-0)^2+(z-0)^2=k^2\left\{(x+a)^2+(y-0)^2+(z-0)^2\right\}$

$\Rightarrow\left(1-k^2\right)\left(x^2+y^2+z^2\right)-2 a x\left(1+k^2\right)+a^2\left(1-k^2\right)=0$

∴ required locus is $x^2+y^2+z^2-\frac{2 a\left(1+k^2\right)}{\left(1-k^2\right)} x+a^2=0$. Which is a sphere.