Locus

Consider a point P (x,y) which is free to move according to certain conditions. The set of points which satisfy the conditions is called the locus of P and the equation which is satisfied by all these points is also called the locus of point P.

 $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)}$  

This formula is used when finding the distance from the point P(x,y) to a given point. 

1. What is the locus of a point which moves so that its distance from the point ( 3, 1) is 2 units.

P (x , y ) - is a variable point
Let A = (3, 1)
PA is the distance from the variable point P to A. (2 units) 
To find equation of locus using points (x,y) and (3,1) : 
 $\therefore PA=2$  

P (x , y) is a variable point
A (3 , 2)
B ( 5 , -1)

The distance PA is equal to the distance PB. To find equation of locus: 
 $\therefore PA=PB$  

P (x , y )
A ( -1 ,2 )
O (0,0)
The distance PA is twice the distance of PO. To find the equation of locus : 

Distance from P to line x = -1 = Distance of P from O
 $\sqrt{\left(x+1\right)^2+\left(y-y\right)^2\ \ }=\sqrt{\left(x-0\right)^2+\left(y-0\right)^2}$  

P(x,y) | Let A = (0,1) | y= -1 = (x, -1)
 Distance from P to A =  Distance from P to y =-1