Question

Верно ли утверждение? 1. Расстояние от точки (2;3) до точки (-3;-2) равно 2корня из 5 2. Середина отрезка, соединяющего точку (4;5) с её проекцией на ось ординат, - это точка (2;5) 3. Прямые у=2х+3 и х=1/2 умножить у + 1/3 4. Расстояние между осями парабол у=х^2+х и у=х^2-х равно 2

Answer 1

1) нет, расстояние равно 5√2
2) да
3) предложение не закончено, но на всякий случай напишу: это две параллельные прямые.
4) нет, расстояние равно 1

Answer 2

1. Расстояние между точками (2,3) и (-3,-2) равно 2√5: To calculate the distance between two points, we can use the distance formula, which is derived from the Pythagorean theorem. The distance between two points (x1, y1) and (x2, y2) is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

In this case, we have (x1, y1) = (2, 3) and (x2, y2) = (-3, -2). Substituting these values into the formula, we get:

d = √((-3 - 2)^2 + (-2 - 3)^2) = √((-5)^2 + (-5)^2) = √(25 + 25) = √50 = 2√5

Therefore, the distance between the points (2,3) and (-3,-2) is indeed 2√5.

2. Середина отрезка, соединяющего точку (4,5) с её проекцией на ось ординат, это точка (2,5): To find the midpoint of a line segment, we can use the midpoint formula. The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by:

(x, y) = ((x1 + x2) / 2, (y1 + y2) / 2)

In this case, we have the point (4,5) and its projection on the y-axis. Since the projection lies on the y-axis, its x-coordinate will be the same as the x-coordinate of the given point (4,5), which is 4. The y-coordinate of the projection will be 0 because it lies on the y-axis.

Substituting the values into the formula, we get:

(x, y) = ((4 + 4) / 2, (5 + 0) / 2) = (8 / 2, 5 / 2) = (4, 2.5)

Therefore, the midpoint of the line segment connecting the point (4,5) with its projection on the y-axis is the point (4, 2.5).

3. Прямые у = 2x + 3 и x = (1/2) * y + (1/3): To find the intersection point of two lines, we can set their equations equal to each other and solve for the variables.

In this case, we have the equations y = 2x + 3 and x = (1/2)y + (1/3). Setting them equal to each other:

2x + 3 = (1/2)y + (1/3)

To solve for x, we can isolate it by subtracting (1/2)y and (1/3) from both sides:

2x - (1/2)y = -3 + (1/3) 4x - y = -9/3 + 1/3 4x - y = -8/3

We now have an equation in terms of x and y. This equation represents the line of intersection between the two given lines.

Therefore, the statement "Прямые у = 2х + 3 и х = 1/2 * у + 1/3" is not accurate. The equation 4x - y = -8/3 represents the line of intersection between the two given lines.

4. Расстояние между осями парабол у = х^2 + х и у = х^2 - х равно 2: To find the distance between the axes of the parabolas y = x^2 + x and y = x^2 - x, we can consider the difference in their y-intercepts.

The y-intercept of the parabola y = x^2 + x is the value of y when x is equal to 0. Substituting x = 0 into the equation, we get:

y = (0)^2 + (0) = 0

Therefore, the y-intercept of y = x^2 + x is 0.

Similarly, the y-intercept of the parabola y = x^2 - x is the value of y when x is equal to 0. Substituting x = 0 into the equation, we get:

y = (0)^2 - (0) = 0

Therefore, the y-intercept of y = x^2 - x is also 0.

Since the y-intercepts of both parabolas are the same, the distance between their axes is 0. Hence, the statement "Расстояние между осями парабол у = х^2 + х и у = х^2 - х равно 2" is not true. The distance between the axes of these parabolas is 0, not 2.