St#2 Circles (Part 1)

The distance from the fixed point to any point on the circle is called the radius. The radius is the distance from the center of the circle to any point on its circumference. It is a fixed distance and is the same for all points on the circle. The other options, such as chord, diameter, and tangent line, do not represent the distance from the fixed point to any point on the circle.

A circle is defined as the set of points in a plane that are equidistant from a fixed point, known as the center. The distance between any point on the circle and the center is always the same, which makes it a constant. Therefore, a circle fits the given description.

The equation x^2 + y^2 = 81 represents a circle centered at the origin (0,0) with a radius of 9. This is because the equation is in the standard form of a circle, where (h,k) represents the center of the circle and r represents the radius. In this case, h = 0, k = 0, and r = 9.

The equation of a circle in standard form with the center at the origin is x^2 + y^2 = r^2. This equation represents all the points on the circle with radius r centered at the origin (0,0). The equation x^2 + y^2 = r represents a circle with radius √r, not r. The equations x^2 + y^2 - r = 0 and x^2 + y^2 - r^2 = 0 do not represent circles.

The fixed point of the circle is called the center. This is because the center is the point from which all other points on the circle are equidistant. It is the point that defines the shape and size of the circle. The center is also important in determining other properties of the circle, such as the radius and diameter.

The equation of a circle is given by x^2 + y^2 = r^2, where (x, y) represents the coordinates of any point on the circle and r represents the radius of the circle. In this case, the equation x^2 + y^2 = 9 represents a circle with a radius of 3 units.

The equation of the circle is given in the form (x-a)^2 + (y-b)^2 = r^2, where (a,b) represents the center of the circle. In this case, the equation 3x^2 + 3y^2 = 12 can be rewritten as (x-0)^2 + (y-0)^2 = (2)^2. This means that the center of the circle is at (0,0), which is the correct answer.

The equation of the circle is given by (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius. In this case, the equation (x - 2)^2 + (y + 1)^2 = 9 represents a circle with its center at (2, -1) and a radius of 3.

The equation of the circle can be rewritten as (x+1)^2 + (y-2)^2 = 25, which is in the standard form (x-a)^2 + (y-b)^2 = r^2. Comparing the equation with the standard form, we can see that the center of the circle is (-1, 2) and the radius is 5.

The equation in standard form of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius. In this question, the center of the circle is (2, -3) and the diameter is 10. Since the diameter is twice the radius, the radius is 10/2 = 5. Plugging in the values, we get (x - 2)^2 + (y + 3)^2 = 5^2 = 25. Therefore, the correct answer is (x - 2)^2 + (y + 3)^2 = 25.

The given equation of the circle is in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. By comparing the given equation (x + 5)^2 + (y - 6)^2 = 4 with the general form, we can see that the center of the circle is (-5, 6) and the radius is 2. The correct answer x^2 + y^2 + 10x - 12y + 57 = 0 is the expanded form of the given equation, which represents the equation of the circle.

The equation of a circle with center at the origin (0,0) and radius √10 is x^2 + y^2 = 10. This equation represents all the points (x, y) that are at a distance of √10 units from the origin.

The given equation of the circle is in the form (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle. By rearranging the given equation, we can rewrite it as (x + 1)2 + (y - 2)2 = 25. Comparing this with the standard form, we can see that the center of the circle is (-1, 2). Therefore, the correct answer is (-1, 2).

The equation in general form of a circle with center (-4, 5) and radius 4 is x^2 + y^2 + 8x - 10y + 25 = 0. This equation represents a circle with center (-4, 5) and radius 4, as the terms x^2 and y^2 represent the squared distances from any point (x, y) on the circle to the center (-4, 5). The coefficients 8 and -10 represent the x and y coordinates of the center multiplied by 2, respectively, while the constant term 25 represents the square of the radius.

The radius of the circle can be found by measuring the distance from the center of the circle to the x-axis. In this case, the center of the circle is at (-4, -1). Since the circle is tangent to the x-axis, the distance from the center to the x-axis is equal to the radius of the circle. Therefore, the radius is 1.