Problem 51 - Precalculus Essentials

Question

Write the standard form of the equation of the circle with the given center and radius. $$(x+1)^{2}+y^{2}=25$$

Step-by-Step Solution

Answer

The equation \((x+1)^{2}+y^{2}=25\) represents a circle with center at (-1,0) and radius of 5.

Step 1: Identify the circle equation

Rewriting the given equation we get \( (x+1)^2 + y^2 = 25 \). From the form of this equation, we can see that it corresponds to the standard form of the circle equation \((x - h)^2 + (y - k)^2 = r^2\).

Step 2: Identify the center

From the given equation equation, we compare \((x - h)^2\) with \((x +1)^2\), we can see that \(h = -1\) since \(-(-1) = +1\). Also, comparing \((y - k)^2\) with \(y^2\), it can be noticed that \(k = 0\) as \(-0 = 0\). Therefore, the center of the circle is \((-1, 0)\).

Step 3 Identify the radius

Looking at the equation, we can see that the radius squared, \(r^2\), is equal to 25 by comparing \(25 = r^2\). Taking the square root of both sides, we determine that the radius, \(r\), is 5.

Key Concepts

Circle Equation

Understanding the equation of a circle is fundamental in geometry. The standard form of a circle's equation is \[ (x - h)^2 + (y - k)^2 = r^2 \], where \( (x, y) \) represents any point on the circle, \( h \) and \( k \) are the coordinates of the center, and \( r \) is the radius. To clarify, \( x \) and \( y \) are variables, while \( h \) and \( k \) are constants that determine the position of the circle within the coordinate plane, and \( r \) is a constant that defines the circle’s size.

When decoding a circle equation, comparing it to the standard form quickly reveals crucial information about the circle's geometry. For example, by rewriting the given equation \( (x+1)^2 + y^2 = 25 \) in standard form, we understand that the circle has a radius of 5 units and is centered at the point (-1,0). This is because the equation matches the standard form with \( h = -1 \) and \( k = 0 \) indicating the circle's center, and \( r^2 = 25 \) signifying a radius of 5 units once we take the square root of 25.

Radius of a Circle

The radius of a circle is the distance from the center of the circle to any point on the circle's perimeter. It is a crucial concept because it defines the size of the circle. In a standard form equation, the radius is represented by \( r \) and is always a positive value. The radius is squared in the circle equation, corresponding to \( r^2 \) in the formula \[ (x - h)^2 + (y - k)^2 = r^2 \].

In the given exercise, identifying the radius is as straightforward as looking at the constant value on the equation's right side. Once the equation \( (x+1)^2 + y^2 = 25 \) is established, we can infer that \( r^2 = 25 \) from which calculating the radius \( r \) requires taking the square root of 25, resulting in \( r = 5 \). Remember, the radius's real-world interpretation is always the non-negative square root, as distances cannot be negative.

Center of a Circle

The center of a circle is the fixed point from which every point on the perimeter is equidistant. In other words, the radius distance is consistent from the center to any point along the circle's edge. The coordinates of the center are signified by \( h \) and \( k \) in the standard form circle equation \[ (x - h)^2 + (y - k)^2 = r^2 \].

When interpreting a circle's equation, the terms \( (x - h)^2 \) and \( (y - k)^2 \) within the equation hint at the coordinates of the center. For instance, the given circle equation \( (x+1)^2 + y^2 = 25 \) implies a center at (-1, 0) because translating \( (x - h)^2 \) to \( (x + 1)^2 \) means \( h \) must be -1, and no \( y \) term implicates that \( k \) is 0. Locating the center is like uncovering the 'home base' of the circle on the coordinate grid, where every point on the circle maintains a steadfast distance, equal to the radius.