Question

Find the standard form of the equation of each circle. Center (-5,6) and area \(49 \pi\)

Short answer

(x + 5)^2 + (y - 6)^2 = 49

Step-by-step solution

Write the General Form of a Circle's Equation

The general form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius.

Identify the Center

From the problem, the center of the circle is given as \( (-5, 6) \). This means \( h = -5 \) and \( k = 6 \).

Calculate the Radius

The area of the circle is given as \( 49 \pi \). The area of a circle is \( \pi r^2 \). Therefore, \( r^2 = 49 \pi / \pi = 49 \). So, \( r = \sqrt{49} =7 \).

Substitute Values into the Circle Equation

Substitute \( h = -5 \, k = 6 \) and \( r = 7 \) into the general form of the circle's equation: \( (x + 5)^2 + (y - 6)^2 = 7^2 \).

Simplify the Equation

Simplify the equation: \( (x + 5)^2 + (y - 6)^2 = 49 \).

Key concepts

circle's general form

The general form of a circle's equation is fundamental to understanding circle equations. It is given by the equation (x - h)^2 + (y - k)^2 = r^2 , where: - h is the x-coordinate of the circle's center, - k is the y-coordinate of the circle's center, and - r is the radius of the circle. This equation states that the distance from any point (x, y) on the circle to the center (h, k) is always the same, which is exactly the radius . By substituting the values of h, k, and r into this general form, you can describe any specific circle.

center of a circle

The center of a circle is an essential concept in understanding circle equations. It is the point from which every point on the perimeter of the circle is equidistant. The general form of the circle's equation, (x - h)^2 + (y - k)^2 = r^2 , shows us that the coordinates (h, k) represent the center of the circle. For example, in the exercise, we are given the center (-5, 6) . Here, h = -5 and k = 6 . By identifying these values, you set the position of the circle in the coordinate plane.

radius calculation

The radius of a circle is the distance from the center to any point on the circle. To calculate the radius when given the area of the circle, you can use the formula for the area: Area = πr^2 . For instance, if the area of the circle is given as 49π , we can set up the equation to find r: πr^2=49π . Dividing both sides by π, we get r^2 = 49 . Taking the square root of both sides, r = 7 . Thus, the radius is 7 . This demonstrates how to determine the radius , given the area.

standard form of a circle's equation

The standard form of a circle's equation provides a clear and straightforward way to describe a circle on the coordinate plane. Once you have the circle's center and radius, you can substitute these values into the general form (x - h)^2 + (y - k)^2 = r^2 to get the circle's specific equation. Using the exercise's values: h = -5, k = 6, and r = 7, substitute these into the equation: (x + 5)^2 + (y - 6)^2 = 7^2 . Simplifying further gives us: (x + 5)^2 + (y - 6)^2 = 49 . This is the standard form of the equation of the circle in the exercise. With these calculations, you can now graph and understand the circle's relationship to the rest of the coordinate plane.