Question

В плоскости xy найдите точку F (x,y,0) равноудалённую от 3 данных точек M (1,1,0) N (0,-1,0) K (-1,0,1) СРОЧНО!!! И ПОЖАЛУЙСТА С ПОДРОБНОСТЯМИ ЧТОБЫ Я САМА ПОНЯЛА

Answer 1

У точек в плоскости xy координата z = 0
Расстояния от точки F(x,y,0) до трёх точек, точнее, квадраты расстояний
r₁² = (x-1)² + (y-1)²
r₂² = x² + (y+1)²
r₃² = (x+1)² + y² + 1²
Все три расстояния равны, как и их квадраты
r₁² = r₂² = r₃² = r²
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r² = x² - 2x + 1 + y² - 2y + 1
r² = x² + y² + 2y + 1
r² = x² + 2x + 1 + y² + 1
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вычтем из первого третье
0 = - 4x - 2y
y = -2x
Подставляем 
r² = x² - 2x + 1 + 4x² + 4x + 1
r² = x² + 4x² - 4x + 1
r² = x² + 2x + 1 + 4x² + 1
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r² = 5x² + 2x + 2 
r² = 5x² - 4x + 1
r² = 5x² + 2x + 2
первое и третье дублируют друг друга, что логично, ведь из них мы и выражали y
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r² = 5x² + 2x + 2 
r² = 5x² - 4x + 1
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вычтем из первого второе
0 = 2x + 2 + 4x - 1
0 = 6x + 1
x = -1/6
y = -2x = 1/3
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Ответ
F (-1/6,1/3,0) 

Answer 2

Finding the point F in the xy-plane equidistant from three given points M, N, and K

To find the point F in the xy-plane that is equidistant from three given points M(1, 1, 0), N(0, -1, 0), and K(-1, 0, 1), we can use the concept of perpendicular bisectors.

The perpendicular bisector of a line segment is a line that is perpendicular to the line segment and passes through its midpoint. In this case, we need to find the perpendicular bisectors of the line segments MN, NK, and KM.

Let's find the equations of the perpendicular bisectors for each line segment:

1. Perpendicular bisector of MN: - Midpoint of MN: ((1 + 0) / 2, (1 + (-1)) / 2, (0 + 0) / 2) = (0.5, 0, 0) - Slope of MN: (1 - (-1)) / (1 - 0) = 2 - Negative reciprocal of the slope: -1/2 - Equation of the perpendicular bisector: y - 0 = (-1/2)(x - 0.5)

2. Perpendicular bisector of NK: - Midpoint of NK: ((0 + (-1)) / 2, (-1 + 0) / 2, (0 + 1) / 2) = (-0.5, -0.5, 0.5) - Slope of NK: (0 - (-1)) / (-1 - 0) = 1 - Negative reciprocal of the slope: -1 - Equation of the perpendicular bisector: y - (-0.5) = (-1)(x - (-0.5))

3. Perpendicular bisector of KM: - Midpoint of KM: ((-1 + 1) / 2, (0 + 1) / 2, (1 + 0) / 2) = (0, 0.5, 0.5) - Slope of KM: (0 - 1) / (1 - (-1)) = -1/2 - Negative reciprocal of the slope: 2 - Equation of the perpendicular bisector: y - 0.5 = 2(x - 0)

Now, we need to find the intersection point of these three perpendicular bisectors. To do that, we can solve the system of equations formed by the three equations of the perpendicular bisectors.

Solving the system of equations: 1. y - 0 = (-1/2)(x - 0.5) 2. y - (-0.5) = (-1)(x - (-0.5)) 3. y - 0.5 = 2(x - 0)

Simplifying the equations: 1. y = (-1/2)x + 0.25 2. y + 0.5 = -x - 0.5 3. y = 2x - 1

Now, we can solve the system of equations by substitution or elimination method to find the coordinates of the point F.

Substituting equation 3 into equation 1: 2x - 1 = (-1/2)x + 0.25

Simplifying and solving for x: (5/2)x = 1.25 x = 1.25 * (2/5) x = 0.5

Substituting the value of x into equation 3: y = 2(0.5) - 1 y = 1 - 1 y = 0

Therefore, the coordinates of the point F are (0.5, 0, 0).

Note: The above calculations were done in the xy-plane, where the z-coordinate is 0.