On the abscissa, find point K equidistant from points A (4; 1) B (-3; 2).

We have two points: A (4; 1) and B (-3; 2).

Let’s find the coordinates of the point K, lying on the abscissa axis, equidistant from the required points.

First, note that the point on the X axis has coordinates (x; 0).

Find the distances AK and BK, and equate them.

AK = ((x – 4) ^ 2 + (0 – 1) ^ 2) ^ (1/2) = (1 + (x – 4) ^ 2) ^ (1/2);

BK = ((x + 3) ^ 2 + (0 – 2) ^ 2) ^ (1/2) = (4 + (x + 3) ^ 2) ^ (1/2);

If the numbers are equal, then their squares are also equal:

1 + (x – 4) ^ 2 = 4 + (x + 3) ^ 2;

1 + x ^ 2 – 8 * x + 16 = 4 + x ^ 2 + 6 * x + 9;

14 * x = 4;

x = 2/7.

(2/7; 0) – coordinates of point K.