Question

The locus of a point which moves so that the difference of the squares of its distances from two given points is constant, is a

• A. straight line
• B. plane
• C. sphere
• D. none of these

Answer 1

Answer: (B)

plane

Let the two fixed points be $(x_1,y_1)$ and $(x_2,y_2)$
Given that ${(x-x_1)}^2+{(y-y_1)}^2+{(z-z_1)}^2-[{(x-x_2)}^2+{(y-y_2)}^2+{(z-z_2)}^2]=k$
$\Rightarrow 2x(x_1+x_2)+2y(y_1+y_2)+2z(z_1+z_2)+k-({x_1}^2+{y_1}^2+{z_1}^2+{x_2}^2+{y_2}^2+{z_2}^2)=0$
Therefore, locus is plane.
Ans: B