# Find the coordinates of a point belonging to the abscissa axis and equidistant from points A (-2; 3) and B (6; 1).

Since the required point must belong to the abscissa axis, its ordinate is zero.

Let’s denote the abscissa of the required point by x.

Thus, the coordinates of a given point (x; 0).

The distance between the desired point and point A is equal to the square root of the sum of the squares of the differences from the coordinates:

√ ((- 2 – x) ^ 2 + (3 – 0)) ^ 2 = √ ((x + 2) ^ 2 + 9).

Similarly, the distance between the desired point and point B:

√ ((6 – x) ^ 2 + (1 + 0)) = √ ((6 – x) ^ 2 + 1).

Since the required point is equidistant from A and B, then:

√ ((x + 2) ^ 2 + 9) = √ ((6 – x) ^ 2 + 1);

x ^ 2 + 4x + 4 + 9 = 36 – 12x + x ^ 2 + 1;

2x ^ 2 + 16x – 24 = 0;

x ^ 2 + 8x + 12 = 0;

D = 8 * 8 – 4 * 1 * 12 = 16;

x1 = (- 8 – √16) / 2 = – 6;

x2 = (- 8 + √16) / 2 = – 2.

Answer: (- 6; 0) and (- 2; 0).