A point moves such that the sum of the squares of its distances from the two sides of length 'a' of a rectangle is twice the sum of the squares of its distances from the other two sides of length 'b'. The locus of the point can be :

Step-by-step solution

Let the two sides of the rectangle lie along x -axis & y axis as shown Given that aligned & k^2+(k-b)^2=2 (h^2+(a-h)^2 ) & 2 k^2-2 k b+b^2=4 h^2-4 a h+2 a^2 aligned Replacing h by x and k by y aligned & 2 y^2-2 b y+b^2=4 x^2-4 a x+2 a^2 & 2 (y^2-b y )+b^2=4 (x^2-a x )+2 a^2 & 2 (y- b 2 )^2+ b^2 2 =4 (x- a 2 )^2+a^2 & 4 (x- a 2 )^2-2 (y- b 2 )^2= b^2 2 -a^2 aligned Hence it is a hyperbola or pair of lines if b^2 2 -a^2 0 or b^2 2 -a^2=0 respectively.