Question

Найти множества точек M плоскости абсолютная величина разности квадратов расстояний которых до 2-х заданных точек A и B постоянна и равна m > 0 (т.е, |AM^2 - BM^2|=m (примените коорди-­ натный метод, поместив начало координат в середину отрезка АВ и направив ось ОХ по прямой АВ)).

Answer 1

Решение дано в приложении.

Answer 2

Finding the Constant Difference of Squares of Distances

To find the sets of points M on the plane where the absolute difference of the squares of the distances to two given points A and B is constant and equal to m, we can apply the coordinate method by placing the origin at the midpoint of the line segment AB and aligning the x-axis with the line AB.

Let's proceed with the solution.

Solution:

Given: - Two points A and B in the plane. - Constant m, where |AM^2 - BM^2| = m.

To solve this problem, we can use the following steps:

1. Find the Midpoint of AB: - Let the coordinates of point A be (x1, y1) and the coordinates of point B be (x2, y2). - The midpoint of AB is given by the coordinates: ((x1 + x2)/2, (y1 + y2)/2).

2. Align the x-axis with AB: - The line AB can be represented by the equation y = mx + c, where m is the slope of AB and c is the y-intercept.

3. Express the Points M in Terms of Parameter t: - Let the coordinates of point M be (x, y). - Express the coordinates of M in terms of the parameter t.

4. Calculate the Distance AM and BM: - Use the distance formula to calculate the distances AM and BM in terms of t.

5. Formulate the Equation |AM^2 - BM^2| = m: - Substitute the expressions for AM and BM into the equation |AM^2 - BM^2| = m.

6. Solve for the Points M: - Solve the equation obtained in the previous step to find the sets of points M on the plane.

By following these steps, we can determine the sets of points M on the plane where the absolute difference of the squares of the distances to points A and B is constant and equal to m.

If you need further assistance or a detailed walkthrough of the solution steps, feel free to ask!