Problem 41 - Algebra A Combined Function

Question

Write an equation of the circle with the given center and radius. See Example 8. $$ (0,0) ; \sqrt{3} $$

Step-by-Step Solution

Answer

The equation of the circle is \( x^2 + y^2 = 3 \).

Step 1: Understanding the Formula

To write the equation of a circle, we use the standard form of the circle's equation: \[ (x - h)^2 + (y - k)^2 = r^2 \]where \((h, k)\) is the center of the circle, and \(r\) is the radius.

Step 2: Substitute the Center

The center of the circle is given as \((0,0)\). Substitute \(h = 0\) and \(k = 0\) into the standard circle equation:\[ (x - 0)^2 + (y - 0)^2 = r^2 \].Simplifying, we get: \[ x^2 + y^2 = r^2 \].

Step 3: Substitute the Radius

The radius of the circle is given as \(\sqrt{3}\). Substitute \(r = \sqrt{3}\) into the equation:\[ x^2 + y^2 = (\sqrt{3})^2 \].Since \((\sqrt{3})^2 = 3\), the equation becomes:\[ x^2 + y^2 = 3 \].

Step 4: Final Equation of the Circle

Now that we have substituted both the center and the radius, the final equation of the circle is:\[ x^2 + y^2 = 3 \].

Key Concepts

Center of a Circle

In mathematics, the center of a circle is an important concept that refers to the fixed point located exactly in the middle of the circle. The center is denoted in coordinate geometry as

• \((h, k)\) in the general equation where \(h\) and \(k\) are the \(x\) and \(y\) coordinates, respectively.
• It's the point where the radii of the circle meet and is equidistant from any point on the circumference.

To consider the importance of the center in equations, it helps us determine the circle's location within a coordinate plane.
For the given example, the center of the circle is

• \((0, 0)\), often referred to as the origin.

This makes our calculations simpler.When we plug these values into the equation, it indicates a circle centered at the origin of the coordinate plane.

Radius of a Circle

The radius of a circle refers to the distance between the center of the circle and any point on its circumference. It is a crucial measurement that determines the size of the circle. In the standard equation of a circle, the radius is denoted by \(r\).
In the given exercise, the radius is

• \(\sqrt{3}\), which is a numerical value representing the circle's extent.

The radius is always a positive value, denoting the measure of the distance accurately.
This positive value ensures the circle is drawn proportionate to its center.
It contributes significantly to the shape and size of the resulting circle equation.

Standard Form of a Circle

The standard form of a circle's equation is a fundamental algebraic representation that helps in identifying essential characteristics of a circle. Given by the formula:

• \((x - h)^2 + (y - k)^2 = r^2 \).

This formula involves:

• \((h, k)\) as the center of the circle.
• \(r\) as the radius.

The left side of the equation represents the squared distances from the center along both the \(x\) and \(y\) axes.
The right side, \(r^2\), represents the square of the radius.
The purpose of this form is to conveniently display the circle’s equation in geometry; making it easy to understand its position and size in a coordinate system.

Substitution in Equations

Substitution is a widely used method in algebra where specific values are inserted into an equation to find a solution or result. In the context of circle equations:

• You substitute the values for the center \((h, k)\) and radius \(r\) into the standard form.

The process involves integrating the known values into the equation and simplifying it:
For instance, with a circle centered at \((0,0)\) and radius \(\sqrt{3}\), after substitution, we find the equation simplifies to \(x^2 + y^2 = 3\).
This step ensures that we come to an accurate and precise representation of the circle.Using substitution efficiently solves the initial question about the circle’s specific equation, maintaining correctness and understanding.