Find a point which is equidistant from the points A( - 5, 4) and B( - 1, 6). How many such points are there?
Let P (h, k) be the point which is equidistant from the points A( - 5, 4) and B( - 1, 6).
By distance formula;
Distance between two points (x1, y1) and (x2, y2);
d = 
PA = PB
(PA)2 = (PB)2
→ ( - 5 - h)2 + (4 - k)2 = ( - 1 - h)2 + (6 - k)2
→25 + h2 + 10h + 16 + k2 - 8k = 1 + h2 + 2h + 36 + k2 – 12k ….
∵ [(a - b)2 = a2 + b2 - 2ab]
→ 25 + 10h + 16 – 8k = 1 + 2h + 36 – 12k
→ 8h + 4k + 41 – 37 = 0
→ 8h + 4k + 4 = 0
→ 2h + k + 1 = 0 ……(i)
Now calculate the,
Mid - point = 
Mid - point of AB = 
At point ( - 3, 5), from Eq. (i);
2h + k = 2( - 3) + 5
= - 6 + 5 = - 1
→ 2h + k + 1 = 0
So, the mid - point of AB satisfies the Eq. (i).
Hence, all points which are solution of the equation 2h + k + 1 = 0 are equidistant from the points A and B.
Replacing h, k by x, y in above equation, we have 2h + k + 1 = 0