Question

Помогите решить, пожалуйста. 1. x²+y²-10y=0 2. x²+y²=6y-4x-4 Нужно найти радиус и координаты центра окружности

Answer 1

1) x^2+y^2-10y+25-25=0
выделяем полный квадрат
x^2+(y-5)^2=25
радиус=5, центр(0;5)
2) x^2+4x+4+y^2-6y=0
(x+2)^2+(y-3)^2=9
радиус=3, центр (-2;3)

Answer 2

Problem Statement

You have two equations: 1. x² + y² - 10y = 0 2. x² + y² = 6y - 4x - 4

You need to find the radius and coordinates of the center of the circle represented by these equations.

Solution

To find the radius and coordinates of the center of the circle, we need to rewrite the equations in the standard form of a circle equation, which is (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r represents the radius.

Let's solve the equations step by step:

Equation 1: x² + y² - 10y = 0

To rewrite this equation in the standard form, we need to complete the square for the y terms. We can do this by adding and subtracting a constant term:

x² + (y² - 10y + 25) = 25

Now, we can rewrite the equation as:

x² + (y - 5)² = 25

Comparing this equation with the standard form, we can see that the center of the circle is at (0, 5) and the radius is √25 = 5.

Equation 2: x² + y² = 6y - 4x - 4

Similarly, let's complete the square for the y terms:

x² + (y² - 6y) = -4x - 4

x² + (y - 3)² = -4x - 4 + 9

x² + (y - 3)² = -4x + 5

Comparing this equation with the standard form, we can see that the center of the circle is at (-2, 3) and the radius is √5.

Therefore, the radius and coordinates of the centers of the circles are as follows:

For Equation 1: - Radius = 5 - Center coordinates = (0, 5)

For Equation 2: - Radius = √5 - Center coordinates = (-2, 3)

Please let me know if you need any further assistance!