Problem 56 - Algebra and Trigonometry

Question

\(55-62\) . Find an equation of the circle that satisfies the given conditions. Center \((-1,-4) ; \quad\) radius 8

Step-by-Step Solution

Answer

The circle's equation is \((x + 1)^2 + (y + 4)^2 = 64\).

Step 1: Understand the Standard Equation of a Circle

The equation of a circle with center \( (h, k) \) and radius \( r \) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] This is an equation you will use to find the equation of the given circle.

Step 2: Identify the Center and Radius from Given Conditions

From the exercise, we know the center of the circle is \((-1, -4)\) and the radius is \(8\). These values correspond to \(h = -1\), \(k = -4\), and \(r = 8\).

Step 3: Substitute the Values into the Circle Equation

Substitute \(h = -1\), \(k = -4\), and \(r = 8\) into the standard circle equation:\[(x - (-1))^2 + (y - (-4))^2 = 8^2\]Simplify to:\[(x + 1)^2 + (y + 4)^2 = 64\]

Step 4: Simplify and Write the Final Equation

The equation is already in its simplest form:\[(x + 1)^2 + (y + 4)^2 = 64\] This is the final equation representing the circle with the specified center and radius.

Key Concepts

Geometry

Understanding circles in geometry involves recognizing their fundamental characteristics like center, radius, and circumference. A circle is a perfectly round shape where all points are equidistant from a single point called the center. This simplicity of geometry is beautiful because it establishes foundational principles used in various applications. Circles can represent the path of celestial bodies or ripples in water.

• **Center:** The core point from which all points on the circle are equally distant.
• **Radius:** The constant distance from the center to any point on the circle.
• **Diameter:** Twice the radius, stretching from one edge of the circle through the center to the opposite edge.
• **Circumference:** The total distance around the circle's edge.

Recognizing these terms helps in solving problems involving circles, such as finding areas, perimeters, or specific equations like the standard equation of a circle.

Circle Center

The circle's center is a vital component, acting as the fixed point for which all points on the circle maintain the same distance called the radius. Given that the center is \((h, k)\), this allows us to anchor the circle in the coordinate plane.
In practical terms, calculating a circle's center often involves:

• Reading directly from given coordinates or data points that determine the central position.
• Locating via an intersection of axes for defined symmetrical shapes.
• Determining position relative to other geometrical figures in complex diagrams.

Remembering that every point on a circle is equidistant from the center can simplify identifying positions and solving equations in circle problems.

Circle Radius

The radius of a circle is constant no matter where you choose to draw it from the center to the edge. This key measurement governs many properties of the circle including its size, area, and perimeter, which are all proportionate to this distance.
In relation to our exercise, the radius is a given value of 8 units. From here, the radius becomes the essential ingredient to integrate into the circle's equation, transforming abstract geometry into concrete numerics.
With a known radius, you can:

• Quickly compute the circle's circumference with the formula \(2\pi r\).
• Determine the area using \(\pi r^2\).
• Establish a boundary for circles when solving integration or optimization problems.

Thus, understanding the radius enhances both basic geometric insight and its application in advanced problems.

Standard Equation

The standard equation for a circle in a Cartesian coordinate plane is written as \((x - h)^2 + (y - k)^2 = r^2\). This formula connects the geometric definition of a circle with algebraic expression.

• **\(h, k\):** The center coordinates making up the circle's fixed point.
• **\(r\):** The radius that must be squared to complete the equation representing the circle's area.

The beauty of this equation is its simplicity and applicability across diverse problems. From calculating the orbits of satellites to plotting perfect curves on a graph, the standard circle equation outlines a precise, mathematically beautiful representation of circular shapes.
In the given example, substituting the center as \((-1,-4)\) and the radius as 8, we arrived at the specific equation: \((x + 1)^2 + (y + 4)^2 = 64\), a clear demonstration of how values transform geometry into an algebraic language.