Question

A(-4;1) B(4;7) AB диаметр Найти: 1. Длину АВ 2. Координаты центра окружности 3. Записать уравнение окружности 2) ABCD параллелограмм А(-2:4) В(-6:12) С(2:8) Найти D

Answer 1

Ответ:

AB(8;6)

|AB|=10

O(0;4)

X^2+(y-4)^2=25

Answer 2

To solve the given problems, let's go step by step:

1. Length of AB: The distance formula can be used to find the length of AB. The distance formula is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Given points A(-4,1) and B(4,7): $AB = \sqrt{(4 - (-4))^2 + (7 - 1)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$

So, the length of AB is 10 units.

2. Coordinates of the center of the circle: Since AB is the diameter of the circle, the center of the circle will lie exactly at the midpoint of AB. The midpoint formula is given by: $\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Using the coordinates of A(-4,1) and B(4,7): $\text{Midpoint} = \left(\frac{-4 + 4}{2}, \frac{1 + 7}{2}\right) = (0, 4)$

So, the coordinates of the center of the circle are (0, 4).

3. Equation of the circle: The general equation of a circle with center (h, k) and radius r is: $(x - h)^2 + (y - k)^2 = r^2$

We have already found the center of the circle as (0, 4). Since AB is the diameter, the radius of the circle is half the length of AB, which is 5 units.

Therefore, the equation of the circle is: $(x - 0)^2 + (y - 4)^2 = 5^2$ $x^2 + (y - 4)^2 = 25$

Now, let's move on to the second problem:

Given points A(-2,4), B(-6,12), and C(2,8) and it is known that ABCD is a parallelogram.

4. Finding point D: A parallelogram has opposite sides parallel. So, the line connecting A and B is parallel to the line connecting C and D. The same applies to the line connecting B and C and the line connecting A and D.

Let's find the vector representing AB and BC:

Vector AB = (x_B - x_A, y_B - y_A) = (-6 - (-2), 12 - 4) = (-4, 8)

Vector BC = (x_C - x_B, y_C - y_B) = (2 - (-6), 8 - 12) = (8, -4)

To find the coordinates of point D, we add vector BC to the coordinates of point A:

Point D(x_D, y_D) = Point A(x_A, y_A) + Vector BC = (-2, 4) + (8, -4) = (6, 0)

So, the coordinates of point D are (6, 0).