For points and of the coordinate plane, a new distance is defined by . Let and . Consider the set of points in the first quadrant which are equidistant (with respect to the new distance) from and 


The set of points consists of 


a) 
One straight line only 

b) 
Union of two line segments 

c) 
Union of two infinite rays 

d) 
Union of a line segment of finite length and an infinite ray 
For points and of the coordinate plane, a new distance is defined by . Let and . Consider the set of points in the first quadrant which are equidistant (with respect to the new distance) from and 


The set of points consists of 


a) 
One straight line only 

b) 
Union of two line segments 

c) 
Union of two infinite rays 

d) 
Union of a line segment of finite length and an infinite ray 
(d)
Let be a general point in the first quadrant such that
(i)
[ and are +ve, point being in first quadrant]
If , then lies in region . Then,
If , then lies in region II. Then,
(not possible)
If then lies in region III. Then,
(not possible)
if , then lies in region IV. Then,
Hence, the required set consists of line segment of finite lengths as shown in the first region and the ray
in the fourth region
Obviously locus of is union of line segment and one infinite ray