Find the coordinates of a point lying in the xy plane and equidistant from points A (0; 1; 0), B (-1; 0; 1), C (0; -1; 0).

| September 18, 2021 | education

To find the coordinates of a point lying in the xy plane (whose applicate z is zero), for example, point K (a; b; 0), equidistant from points A (0; 1; 0), B (-1; 0; 1) , С (0; -1; 0), it is necessary to write formulas for determining the distances from point K to these points and equate these distances. KA ^ 2 = (0 – a) ^ 2 + (1 – b) ^ 2 + (0 – 0) ^ 2 = a ^ 2 + (1 – b) ^ 2; KB ^ 2 = (- 1 – a) ^ 2 + (0 – b) ^ 2 + (1 – 0) ^ 2 = (- 1 – a) ^ 2 + (- b) ^ 2 + 1 ^ 2; KS ^ 2 = (0 – a) ^ 2 + (- 1 – b) ^ 2 + (0 – 0) ^ 2 = (- a) ^ 2 + (- 1 – b) ^ 2. From equations 1 and 3 we obtain the equation: a ^ 2 + (1 – b) ^ 2 = (- a) ^ 2 + (- 1 – b) ^ 2; (1 – b) ^ 2 = (1 + b) ^ 2; 1 – b = 1 + b; b = 0. From equations 1 and 2 we obtain the equation: a ^ 2 + (1 – b) ^ 2 = (- 1 – a) ^ 2 + (- b) ^ 2 + 1 ^ 2; a ^ 2 + (1 – b) ^ 2 = (- 1 – a) ^ 2 + b ^ 2 + 1; substitute b = 0, we get, a ^ 2 + 1 = (- 1 – a) ^ 2 + 1; a = – 1 – a; a = – 0.5.
Answer: the required point K (- 0.5; 0; 0).



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.