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Find the point on the x-axis which is equidistant from $(3,\ -5)$ and $(-2,\ 4)$.
Given: A point on the x-axis which is equidistant from $( 3,\ -5)$ and $( 2,\ -4)$.
To do: To find the point.
Solution:
Let $P( x,\ 0)$ be the point on the x-axis which is equidistant from $A( 3,\ -5)$ and $B( 2,\ -4)$.
$\Rightarrow PA=PB$
On using the distance formula, Distance between the two points $( x_1,\ y_1)$ and $( x_2,\ y_2)=\sqrt{( x_2-x_1)^2+( y_2-y_1)^2}$
$\Rightarrow \sqrt{( 3-x)^2+(-5-0)^2}=\sqrt{( 2-x)^2+( -4-0)^2}$
$\Rightarrow \sqrt{9-6x+x^2+25}=\sqrt{4-4x+x^2+16}$
$\Rightarrow 34-6x+x^2=x^2-4x+20$
$\Rightarrow -6x+4x=20-34$
$\Rightarrow -2x=-14$
$\Rightarrow x=\frac{-14}{-2}=7$
Therefore, point $( 7,\ 0)$ is equidistant from $( 3,\ -5)$ and $( 2,\ -4)$.