Question

Find a point on the X-axis which is equidistant from the points $(5,4)$ and $(-2,3)$.

• A. $(2,0)$
• B. $(0,2)$
• C. $(-2,0)$
• D. $(0-2)$

Answer 1

The correct option is A $(2,0)$

Let the point on the x-axis be $(x,0)$
Distance between $(x,0)$ and $(5,4)=\sqrt{{(5-x)}^2+{(4-0)}^2}=\sqrt{5^2+x^2-10x+16}=\sqrt{x^2-10x+41}$
Distance between $(x,0)$ and $(-2,3)=\sqrt{{(-2-x)}^2+{(3-0)}^2}=\sqrt{2^2+x^2+4x+9}=\sqrt{x^2+4x+13}$

As the point $(x,0)$ is equidistant from the two points, both the distances calculated are equal.

$\sqrt{x^2-10x+41}=\sqrt{x^2+4x+13}$

$=>x^2-10x+41=x^2+4x+13$

$41-13=10x+4x$

$28=14x$
$x=2$. Thus, the point is $(2,0)$.