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Find the point on the x-axis which is equidistant from $( 5,\ -5)$ and $( 0,\ 9)$.
Given: A point on the x-axis which is equidistant from $( 5,\ -5)$ and $( 0,\ 9)$.
To do: To find the point.
Solution:
Let $P( x,\ 0)$ be the point on the x-axis which is equidistant from $A( 5,\ -5)$ and $B( 0,\ 9)$.
$\Rightarrow PA=PB$
On using the distance formula, Distance between the two points $( x_1,\ y_1)$ and $( x_2,\ y_2)=\sqrt{( x_2-x_1)^2+( y_2-y_1)^2}$
$\Rightarrow \sqrt{( 5-x)^2+(-5-0)^2}=\sqrt{( 0-x)^2+( 9-0)^2}$
$\Rightarrow \sqrt{25-50x+x^2+25}=\sqrt{x^2+81}$
$\Rightarrow 50-50x+x^2=x^2+81$
$\Rightarrow 50-50x=81$
$\Rightarrow -50x=81-50$
$\Rightarrow -50x=31$
$\Rightarrow x=-\frac{31}{50}$
Therefore, point $( -\frac{31}{50},\ 0)$ is equidistant from $( 5,\ -5)$ and $( 0,\ 9)$.