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).