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Question
Write the coordinates of a point on X-axis which is equidistant from the points (−3, 4) and (2, 5).
Solution
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) .`
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