Problem 37 - College Algebra

Question

Write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(-3,-1), r=\sqrt{3} $$

Step-by-Step Solution

Answer

The standard form of the equation of the circle is \((x + 3)^{2} + (y + 1)^{2} = 3\).

Step 1: Identify the given values

From the problem, we are given the coordinates of the center of the circle (h, k) = (-3, -1) and radius r = \(\sqrt{3}\).

Step 2: Substitute the given values into the standard form

Substitute h = -3, k = -1, and r = \(\sqrt{3}\) into the standard form \((x - h)^{2} + (y - k)^{2} = r^{2}\). Which gives us \((x - (-3))^{2} + (y - (-1))^{2} = (\(\sqrt{3}\))^{2}\) .

Step 3: Simplify the equation

Simplify the equation to \((x + 3)^{2} + (y + 1)^{2} = 3\). This is the standard form of the equation of the circle.

Key Concepts

Standard Form of a Circle

The equation of a circle in its standard form is an essential concept in geometry, making it easier to work with circles in coordinate planes. This form provides a clear and simple way to define a circle using its center coordinates and radius. The standard form of the equation of a circle is \[ (x - h)^2 + (y - k)^2 = r^2 \], where:

• \( (h, k) \) are the coordinates of the center of the circle.
• \( r \) is the radius, or the distance from the center of the circle to any point on its perimeter.

By expressing a circle in this manner, we can also easily identify its center and radius, which are crucial for graphing or analyzing the circle's properties. Therefore, understanding how to use and manipulate this standard form is fundamental for solving problems related to circles.

Circle Center

The center of a circle is a pivotal point in understanding and describing the circle's location in a coordinate plane. In the equation \[(x - h)^2 + (y - k)^2 = r^2\] of a circle, the center is represented by the coordinates \( (h, k) \). In the given example, these coordinates are \((-3, -1)\), meaning this circle is centered at this point on the Cartesian plane.

• The \( h \) value is the x-coordinate of the center.
• The \( k \) value is the y-coordinate of the center.

Understanding the center is crucial as it helps us determine where the circle is situated relative to the other points or shapes in a geometry problem. It's also the point from which all points at a distance equal to the radius extend outwards to form the perimeter of the circle.

Radius of a Circle

The radius of a circle is one of its most defining attributes, as it measures the distance from the center to any point on its perimeter. When using the standard form equation of a circle \[ (x - h)^2 + (y - k)^2 = r^2 \], \( r \) represents the radius of the circle. In our specific case, the radius is given as \( \sqrt{3} \).

• The radius is always a positive value.
• The radius determines the size of the circle.

This particular circle's radius of \( \sqrt{3} \) means that every point on its edge is \( \sqrt{3} \) units away from its center point \((-3, -1)\). Radii are consistently important in both calculating the area and the circumference of a circle in more advanced geometry and trigonometry problems.

Simplifying Equations

Simplifying equations is a crucial step to make a mathematical expression clearer and more manageable. For the circle example, we started with the form\[(x + 3)^2 + (y + 1)^2 = (\sqrt{3})^2\].Simplifying involves resolving any arithmetic operations and simplifying any expressions like squares or roots that are squared.

• Replacing \( (\sqrt{3})^2 \) with the value 3.
• The expression becomes \( (x + 3)^2 + (y + 1)^2 = 3 \).

This simplification makes the equation easier to interpret and use for further calculations, such as determining the circle's area or analyzing its graph. Simplifying is not just about aesthetically cleaner equations; it also makes mathematical operations more efficient and results easier to understand.