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 :
Text Solution
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(c,d)Let the two sides of the rectangle lie along x-axis & y axis as shown
Given that
(PA) 2 + (PB) 2 = 2 (PC 2 + PD 2 ) ⇒ k 2 + (k – b) 2 = 2 (h 2 + (a – h) 2 )
⇒ 2k 2 – 2kb + b 2 = 4h 2 – 4ah + 2a 2
Replacing h by x and k by y
⇒ 2y 2 – 2by + b 2 = 4x 2 – 4ax + 2a 2 ⇒ 2(y 2 – by) + b 2 = 4(x 2 – ax) + 2a 2
2 
+ 
= 4 
+ a 2 ⇒ 4 
– 2 
= 
– a 2
Hence it is a hyperbola or pair of lines if 
– a 2 ≠ 0 or 
= 0 respectively.
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