Question

Find a point which is equidistant from the points A(–5, 4) and B(–1, 6)? How many such points are there?

Answer 1

Let P(h, k) be the point which is equidistant from the points A(–5, 4) and B(–1, 6).

∴ PA = PB   ...`[∵ "By distance formula, distance" = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)]`

⇒ (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

⇒ 25 + 10h + 16 – 8k = 1 + 2h + 36 – 12k

⇒ 8h + 4k + 41 – 37 = 0

⇒ 8h + 4k + 4 = 0

⇒ 2h + k + 1 = 0   ...(i)

Mid-point of AB = `((-5 - 1)/2, (4 + 6)/2)` = (– 3, 5)   ...`[∵ "Mid-point" = ((x_1 + x_2)/2, (y_1 + y_2)/2)]`

At point (– 3, 5), from equation (i),

2h + k = 2(– 3) + 5

= – 6 + 5

= – 1

⇒ 2h + k + 1 = 0

So, the mid-point of AB satisfy the equation (i).

Hence, infinite number of points, in fact 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 2x + y + 1 = 0