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 (x_{1}, y_{1}) and (x_{2}, y_{2});

d =

PA = PB

(PA)^{2} = (PB)^{2}

→ ( - 5 - h)^{2} + (4 - k)^{2} = ( - 1 - h)^{2} + (6 - k)^{2}

→25 + h^{2} + 10h + 16 + k^{2} - 8k = 1 + h^{2} + 2h + 36 + k^{2} – 12k ….

∵ [(a - b)^{2} = a^{2} + b^{2} - 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

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