Problem 55 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The center of the circle is at the origin (0,0) and the radius of the circle is \(\sqrt{2}\).

Step 1: Identify the center

The equation is already in the standard form i.e., \(x^{2}+y^{2}=r^{2}\). Since there are no constant values added or subtracted with \(x^{2}\) or \(y^{2}\), the center of the circle is at the origin (0,0).

Step 2: Identify the radius

For the standard equation of a circle \(x^{2}+y^{2}=r^{2}\), the right side of the equation gives the square of the radius. In this case, \(r^{2}\) equals 2. To get the radius, take a square root of this value. So, \(r= \sqrt{2}\).

Key Concepts

Center of a Circle

In the equation of a circle, the center is a key component that determines its position on a coordinate plane. The most typical format to find this center is in the equation \[(x-h)^2 + (y-k)^2 = r^2\]Here,

• he center of the circle is \((h, k)\).

If you have an equation like \(x^2 + y^2 = r^2\), which matches the given example \(x^{2}+y^{2}=2\), there are no "h" and "k" values, which means the center is at \((0, 0)\), known as the origin. Understanding this can help one visualize where the circle is plotted even before sketching it.

Radius of a Circle

The radius is the distance from the center of a circle to any point on its circumference. It's crucial in defining the size of a circle, measured as half the diameter. In a circle's equation \((x-h)^2 + (y-k)^2 = r^2\),

• the radius is denoted by \(r\).

In the provided equation\(x^2 + y^2 = 2\), we see that \(r^2 = 2\). To find the radius itself, taking the square root of 2 is necessary, giving us \(r = \sqrt{2}\). This displays how the radius relates directly to the equation of the circle.

Standard Form of a Circle Equation

The standard form of a circle's equation essentially reveals both the circle's center and its radius. This form is\((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the circle's center, and \(r\) is its radius. Understanding this formulation is critical because:

• It instantly allows the center and radius of a circle to be human-readable.
• Simplifies the process of transforming into graphs.

For the equation\(x^2 + y^2 = 2\), it represents a circle with its center at \((0,0)\) and a radius calculated to be \(\sqrt{2}\). MessageLookup the equation will frequently be utilized in geometric and algebraic calculations.

Square Root

The square root is a mathematical operation used to find a number that, when multiplied by itself, yields the original number. For a circle equation like \(x^2 + y^2 = r^2\), finding the radius involves taking the square root of \(r^2\). For \(x^2 + y^2 = 2\), we need \(\sqrt{2}\) to determine the radius.

• Remember, \(\sqrt{a^2} = a\), so you are effectively "undoing" the squaring process.
• With non-perfect squares like 2, the square root can't be simplified into an exact integer or simple fraction.

The square root operation provides a direct bridge to understanding dimensions in Euclidean space.

Origin

The origin in a coordinate plane is the point \((0, 0)\). It acts as the central reference point where both the x-axis and y-axis intersect. When a circle's equation simplifies to \(x^2 + y^2 = r^2\), the absence of additional terms involving x and y denotes that:

• The circle is centered at the origin.
• Coordinates stand essentially zeroed around the point \((0, 0)\).

Circles centered at the origin make calculating distances, like the radius, straightforward since there's no horizontal or vertical displacement to consider. This foundational knowledge is essential for students dealing with geometric transformations and symmetries.