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:

Options

A. a circle
B. an ellipse
C. a hyperbola
D. a pair of lines

Answer

C, D

Explanation

Let the two sides of the rectangle lie along x-axis and y-axis.

Let point P be (h,k)
Given that
`(PA)^(2) + (PB)^(2) = 2 (PC^(2)+PD^(2))`
`rArrk^(2) - (k-b)6(2) = 2(h^(2) + (a-h)^(2))`
`rArr 2k^(2) - 2kb + b^(2) = 4h^(2) - 4ah + 2a^(2)`
Replacing h by x and k by y
`rArr 2y^(2) - 2by + b^(2) = 4x^(2) - 4ax + 2a^(2)`
or `2(y^(2)-by) + b^(2) = 4(x^(2)-ax) + 2a^(2)`
or `4(x-(a)/(2))^(2) - 2(y-(b)/(2))^(2) = (b^(2))/(2) -a^(2)`
Hence it is a hyperbola if `(b^(2))/(2) -a^(2) ne 0`
or pair of lines if `(b^(2))/(2) -a^(@) =0`