Question

Write the coordinates of a point on x-axis which is equidistant from the points (–3, 4) and (2, 5).

Answer 1

The distance d between two points `(x_1, y_1)` and `(x_2, y_ 2)` is given by the formula

`d =  sqrt((x_1 - x_2)^2 + (y_1 - y_2)^2)`

Here we are to find out a point on the x−axis which is equidistant from both the points

A(−3, 4) and B(2, 5).

Let this point be denoted as C(x, y).

Since the point lies on the x-axis the value of its ordinate will be 0. Or in other words we have y = 0.

Now let us find out the distances from ‘A’ and ‘B’ to ‘C’

`Ac = sqrt((- 3 - x)^2 + (4 - y)^2)`

`= sqrt((-3 - x)^2 + (4 - 0)^2)`

`AC = sqrt((-3 - x)^2 + (4)^2`

`BC= sqrt((2 - x)^2 + (5 - y)^2)`

`= sqrt((2 - x)^2 + (5 - 0)^2)`

`BC = sqrt((2 - x)^2 + (5)^2)`

We know that both these distances are the same. So equating both these we get,

AC = BC 

`sqrt((-3-x)^2 + (4)^2) = sqrt((2 - x)^2 + (5)^2)`

Squaring on both sides we have,

`(-3 - x)^2 + (4)^2 = (2 - x)^2 + (5)^2`

`9 + x^2 + 6x + 16 = 4 + x^2 - 4x + 25`

`10x = 4`

`x = 2/5`

Hence the point on the x-axis which lies at equal distances from the mentioned points is `(2/5, 0)`.