Problem 45 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recall the Standard Form of a Circle's Equation

The standard form of a circle's equation with center at \(h, k\) and radius \mathit{r}\ is given by: \( (x - h)^2 + (y - k)^2 = r^2 \). Here, the center and radius are provided.

Step 2: Substitute the Center Coordinates into the Equation

The center of the circle is \(0, 0\). Substitute \ h = 0 \ and \ k = 0 \ into the equation, resulting in: \( (x - 0)^2 + (y - 0)^2 = r^2 \), which simplifies to \( x^2 + y^2 = r^2 \).

Step 3: Substitute the Radius into the Equation

The radius given is \ \sqrt{3} \, so substitute \ r = \sqrt{3} \ into the equation \( x^2 + y^2 = r^2 \). Hence, \( x^2 + y^2 = (\sqrt{3})^2 \) which simplifies to \( x^2 + y^2 = 3 \).

Step 4: Write the Final Equation of the Circle

After substituting the given center and radius into the standard form, the equation of the circle is \( x^2 + y^2 = 3 \).

Key Concepts

Center of a Circle

The center of a circle is a crucial concept in understanding circle equations. It refers to the fixed point that is equidistant from all points on the circle. In the context of equations, the center is simply the coordinate pair \(h, k\).
When discussing circle equations, the center provides a reference to position the circle on a coordinate plane.

• If the center is at \(0, 0\), like in our example, it means the circle is centered at the origin of the coordinate system.
• For circles not centered at the origin, you'd use the coordinates \(h, k\) in the equation format.

Understanding the center helps in visualizing where the circle is located and plays a vital role when substituting into the standard form of a circle's equation.

Radius of a Circle

The radius of a circle is defined as the constant distance from the center of the circle to any point on its circumference. In mathematical terms, it's represented by the variable \(r\).
The radius is a key component in defining the size of the circle.

• A larger radius means a larger circle, as each point on the boundary is further from the center.
• Conversely, a smaller radius results in a smaller circle, with the boundary points closer to the center.

For any circle equation, substituting the radius gives you the precise size of that circle. For instance, a given radius of \(\sqrt{3}\) represents a real number that tells you exactly how far each point on the circle is from the center.

Standard Form of a Circle's Equation

The standard form of a circle's equation is a universal format used to express all possible circles on a coordinate plane. The formula is given by: \[(x - h)^2 + (y - k)^2 = r^2\]where \(h\) and \(k\) are the coordinates of the center, and \(r\) is the radius.
This equation form ensures any circular curve can be precisely described.

• The part \((x - h)^2 + (y - k)^2\) ensures symmetry around the center.
• The \(r^2\) part gives the equation its circular characteristics by governing the circle's size.

When the center \(h, k\) is at the origin \(0, 0\), the equation simplifies to \(x^2 + y^2 = r^2\). This form is specifically tailored for circles with centers located on the origin, like our given example, making it easy to interpret and graph.