Question

The coordinates of the point on x-axis which is equidistant from the points $(5,4)$ and $(-2,3)$ are:

• A. $(2,0)$
• B. $(3,0)$
• C. $(0,2)$
• D. $(0,3)$

Answer 1

Answer: (A)

$(2,0)$

Distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ can be calculated using the formula $\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

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.

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

squaring on both the sides

$\Rightarrow x^2-10x+41=x^2+4x+13$

$41-13=10x+4x$

$28=14x$

$x=2$

Thus, the point is $(2,0)$