Problem 57 - Algebra 2

Question

Find the center and the radius of each circle. $$ x^{2}+y^{2}=14 $$

Step-by-Step Solution

Answer

The center of the circle is at the origin, (0,0), and the radius of the circle is \( \sqrt{14}\).

Step 1: Identify the form of the equation

The given equation is \(x^{2}+y^{2}=14\). This is already in the standard form of a circle, \(x^{2}+y^{2}=r^{2}\), where the values of x and y are squared and added together, which equals to the square of radius r.

Step 2: Identify the Center

As the equation is already in standard form, and there's no number subtracted or added to the x and y terms (like \(x-h\) or \(y-k\)), we can see that the circle is centered at the origin, which means the center is at (0,0).

Step 3: Identify the Radius

The square of the radius (denoted as r) equals 14 in the given circle equation. To find the radius, we just take the square root of 14. Hence, the radius r of the circle is \( \sqrt{14}\).

Key Concepts

Center of a Circle

When it comes to understanding the equation of a circle, knowing how to find the center is essential. The center is represented in the equation by the point \(h, k\), where \(h\) and \(k\) are the values that shift x and y horizontally and vertically from the origin. In its simplest form, the equation \(x^2 + y^2 = r^2\) suggests that both x and y are squared.

This implies the circle has a center at the origin since there are no additional terms like \(x - h\) or \(y - k\). So, when you see \(x^2 + y^2\) in a circle equation, you can immediately deduce that the circle's center is at (0, 0). Simplifying complex equations to this form is a fundamental skill when dealing with circle equations.

Radius of a Circle

The radius of a circle is an inherent part of its equation and determines the size of the circle. When a circle equation is given in the form \(x^2 + y^2 = r^2\), determining the radius is straightforward.

The term on the right side of the equation, \(r^2\), denotes the square of the radius. Therefore, to find the actual radius, you need to take the square root of that term.

• For example, if the equation is \(x^2 + y^2 = 14\), you find the radius by calculating \(\sqrt{14}\).

Comprehending how to extract the radius from an equation in standard form is vital for understanding the circle's properties and dimensions.

Standard Form of a Circle Equation

Understanding the standard form of a circle equation is key to analyzing any circular graph quickly. The standard form is expressed as \( (x - h)^2 + (y - k)^2 = r^2 \), where:

• \(h\) and \(k\) are the coordinates of the circle's center.
• \(r\) is the radius.

When no numbers are subtracted from x or y, like in \(x^2 + y^2 = r^2\), it indicates that the circle is centered at (0, 0).

This simplification helps students interpret and plot circles even if the values for \(h\) and \(k\) are not explicitly present. Moreover, recognizing this format enables easy transformations and analysis of shifts and changes in circle equations.