Problem 5 - Precalculus

Question

The equation \((x-h)^{2}+(y-k)^{2}=r^{2}\) is the standard form of the equation of a __________ with center __________ and radius ____________.

Step-by-Step Solution

Answer

The equation represents a circle with the center at the point (h, k) and the radius is r.

Step 1: Identifying the Type of Geometric Figure

Looking at the given equation, one can recognize the standard form of the equation of a circle.

Step 2: Identifying the Center of the Geometric Figure

The center of the circle in an equation of this form is given by the values h, k, which are subtracted from x, y respectively. Therefore, the center of the circle in this equation would be represented by the point (h, k).

Step 3: Identifying the Radius of the Geometric Figure

The value of r in this type of equation actually represents the radius of the circle.

Key Concepts

standard form

The standard form of a circle's equation is fundamental to understanding its geometry. This form is expressed as \[(x-h)^2 + (y-k)^2 = r^2\]where:

• \((x, y)\) are the coordinates of any point on the circle.
• \(h\) and \(k\) are the coordinates of the circle's center.
• \(r\) is the radius of the circle.

Recognizing this equation is key to differentiating circles from other geometric figures, such as parabolas or ellipses, which have different standard forms. By using the standard form, you can retrieve detailed information about the circle's properties through simple exploration of its components. Each segment of the equation directly leads to understanding either the center or the radius, simplifying your problem-solving process.

center of circle

The center of a circle is a crucial element found within its equation's standard form. It is represented by the point \((h, k)\) in the equation \[(x-h)^2 + (y-k)^2 = r^2\].

• \(h\) is the x-coordinate of the circle's center.
• \(k\) is the y-coordinate of the circle's center.

When identifying the circle's center from this equation, recognize that the values \(h\) and \(k\) are "hiding" inside the expressions \((x-h)^2\) and \((y-k)^2\). Remember that both coordinates are subtracted from \(x\) and \(y\) in the formula, thus making it essential to carefully observe these signs. If you see a plus sign in the equation, remember it inversely affects the perceived center. For instance, \((x+2)^2\) points to a center at \((-2, k)\). Successfully identifying the center is a first step to graphing or understanding a circle's position on a coordinate plane.

radius of circle

The radius of a circle in the context of its equation is the distance from the center to any point on the circle itself. In the standard form equation \[(x-h)^2 + (y-k)^2 = r^2\], \(r\) represents this radius. However, it appears as \(r^2\) in the equation. Therefore, to find the actual radius, you must take the square root of the value on the right-hand side of the equation. For example, if \[r^2 = 25\], then \(r = 5\), because \(\sqrt{25} = 5\). This value tells you how far the circle extends from its center, providing insight into its size. It is important to remember that the radius is always a positive distance, reflecting the real-world measurement from center to perimeter.