Find a point which is equidistant from the points A (-5, 4) and B (-1,

1.	Find a point which is equidistant from the points A (-5, 4) and B (-1, 6). How many such points are there?
Answer» Let P(x, y) be the equidistant point from points A (-5, 4) and B (-1, 6). So, the mid-point can be the required point (x, y) = ( (-5 – 1/ 2), (4 + 6)/2 ) (x , y) = (-6/2 , 10/2) = (-3, 5) Thus, the required point is (-3, 5) Now, We also know that, AP = BP So, AP² = BP² (x + 5)² + (y – 4)² = (x + 1)² + (y – 6)² x² + 25 + 10 + y² – 8y + 16 = x² + 2x + 1 + y² – 12y + 36 10x + 41 – 8y = 2x + 37 – 12y 8x + 4y + 4 = 0 2x + y + 1 = 0 Therefore, all the points which lie on the line 2x + y + 1 = 0 are equidistant from A and B.

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

Answer»

Let P(x, y) be the equidistant point from points A (-5, 4) and B (-1, 6).

So, the mid-point can be the required point

(x, y) = ( (-5 – 1/ 2), (4 + 6)/2 )

(x , y) = (-6/2 , 10/2) = (-3, 5)

Thus, the required point is (-3, 5)

Now,

We also know that, AP = BP

So, AP² = BP²

(x + 5)² + (y – 4)² = (x + 1)² + (y – 6)²

x² + 25 + 10 + y² – 8y + 16 = x² + 2x + 1 + y² – 12y + 36

10x + 41 – 8y = 2x + 37 – 12y

8x + 4y + 4 = 0

2x + y + 1 = 0

Therefore, all the points which lie on the line 2x + y + 1 = 0 are equidistant from A and B.

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