Let A, B and C be three points in a plane. The locus of a point P moving such that P A^2+P B^2=2 P C^2 is a

Step-by-step solution

Let A (x₁, y₁ ), B (x₂, y₂ ) and (x₃, y₃ ) and p(x, y) be points on x y plane. Now, P A^2+P B^2=2 P C^2 gathered (x-x₁ )^2+ (y-y₁ )^2+ (x-x₂ )^2+ (y-y₂ )^2 =2 [ (x-x₃ )^2+ (y-y₃ )^2 ] x^2+x₁^2-2 x x₁+y^2+y₁^2-2 y y₁ +x^2+x₂^2-2 x x₂+y^2+y₂^2-2 y y₂ =2 [x^2+x₃^2-2 x x₃+y^2+y₃^2-2 y y₃ ] x (-2 x₁-2 x₂+4 x₃ )+y (-2 y₁-2 y₂+4 y₃ ) +x₁^2+x₂^2+y₁^2+y^2-2 x₃^2-2 y₃^2=0 gathered this equation is in the form of a x+b y+c=0 Hence, locus of p is a straight line.