Question

Скласти рівняння кола, яке проходить через точки А(2; 2), В(– 4; 2), якщо його центр лежить на прямій у + х + 2 = 0.

Answer 1

Equation of a Circle Passing Through Two Points

To find the equation of a circle passing through points A(2, 2) and B(-4, 2), given that its center lies on the line x + 2 = 0, we can follow these steps:

1. Find the midpoint of the line segment AB. This will give us the coordinates of the center of the circle. 2. Find the radius of the circle by calculating the distance between the center and one of the given points. 3. Use the formula for the equation of a circle to write the equation.

Let's go through these steps in detail.

Step 1: Find the Midpoint of AB

To find the midpoint of AB, we can use the midpoint formula:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Substituting the coordinates of A(2, 2) and B(-4, 2) into the formula, we get:

Midpoint = ((2 + (-4)) / 2, (2 + 2) / 2) = (-1, 2)

So, the coordinates of the center of the circle are (-1, 2).

Step 2: Find the Radius

To find the radius of the circle, we can calculate the distance between the center (-1, 2) and one of the given points, A(2, 2). We can use the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Substituting the coordinates of A(2, 2) and the center (-1, 2) into the formula, we get:

Distance = √((2 - (-1))^2 + (2 - 2)^2) = √(3^2 + 0^2) = √9 = 3

So, the radius of the circle is 3.

Step 3: Write the Equation of the Circle

The equation of a circle with center (h, k) and radius r is given by:

(x - h)^2 + (y - k)^2 = r^2

Substituting the coordinates of the center (-1, 2) and the radius 3 into the equation, we get:

(x - (-1))^2 + (y - 2)^2 = 3^2

Simplifying, we have:

(x + 1)^2 + (y - 2)^2 = 9

Therefore, the equation of the circle passing through points A(2, 2) and B(-4, 2), with its center lying on the line x + 2 = 0, is (x + 1)^2 + (y - 2)^2 = 9.

Please let me know if you need any further assistance!