Problem 35 - College Algebra Essentials

Question

write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(-1,4), r=2 $$

Step-by-Step Solution

Answer

The standard form of the equation for the circle with center (-1,4) and radius 2 is \((x + 1)^2 + (y - 4)^2 = 4\).

Step 1: Identify the given values

We are given the center of the circle as (-1,4) and the radius as 2. So, h = -1, k = 4 and r = 2.

Step 2: Substitute the values in the standard form for circle equation

The standard form for a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\). Substituting the given values we get \((x - (-1))^2 + (y - 4)^2 = 2^2\).

Step 3: Simplify the equation

The equation can be simplified to \((x + 1)^2 + (y - 4)^2 = 4\).

Key Concepts

Circle Equation

Understanding the equation of a circle is a fundamental concept intertwining geometry and algebra. A circle's equation may look daunting at a glance, but it follows a simple, elegant format tied closely to the circle's geometrical properties. In algebraic terms, the standard form of a circle's equation is \(x - h)^2 + (y - k)^2 = r^2\), where \(h, k\) are the coordinates of the circle's center and \(r\) is the radius.

The equation essentially captures all points \(x, y\) that lie on the circle's boundary. This equation tells us that if you take any point on the circle, and calculate the square of the horizontal distance from the center \(x - h\) and add it to the square of the vertical distance from the center \(y - k\), it will always equal the square of the radius \(r^2\).

Additionally, to enhance comprehension, one can visualize the circle as a collection of dots that create a perfect loop, equidistant from a central point. The equation is simply a mathematical representation of that visual, encapsulating the essence of circular symmetry in algebraic form.

Geometry in Algebra

The remarkable aspect of algebra is its ability to describe geometric shapes using equations. Geometry in algebra provides a direct link between numerical expressions and geometrical figures, such as lines, circles, and parabolas. This intermingling of disciplines allows us to exploit algebraic techniques to solve geometric problems efficiently and vice versa.

Algebraic Representation of Geometry

For instance, the standard form of a circle's equation is an algebraic expression that reflects every point lying on the periphery of a geometric circle. Likewise, when we plot these algebraic solutions on a coordinate plane, we transition from the abstract realm of numbers to the concrete visual perspective of geometry, manifesting the solutions graphically. Furthermore, this convergence means that the skills you develop in algebra, such as manipulating equations and understanding variables, are directly applicable to solving geometric challenges.

The marriage of algebra and geometry is underpinned by the Cartesian coordinate system, enabling us to navigate through and interpret the spatial characteristics of algebraic equations.

Radius and Center of a Circle

The radius and the center are the most defining elements of a circle. The center is the fixed point from which every point on the circumference is equidistant. In the standard equation of a circle, the center is represented by the ordered pair \(h, k\).

Imagine the center as the heart of the circle, the anchor point from which every line segment, called a radius, extends out to the edge of the circle. The length of these segments is the radius, denoted as \(r\). Since the radius is constant for a given circle, you could say it's the circle's heartbeat — steady, unfaltering, and equal from center to circumference wherever you measure.

The equation of a circle tells us not only the fixed distance from the center to the boundary (the radius) but also locates the circle precisely in the coordinate plane. This precise location and measure mean we have a complete description of the circle: where it sits and how big it is. Converting this into real-world application, if you're planting a circular garden bed with a fixed point for a fountain (the center) and want an even border (the radius), the concept of radius and center gives you the exact blueprint to do so.