Problem 2 - Algebra for College Students

Question

For Problems 1-14, write the equation of each of the circles that satisfies the stated conditions. In some cases there may be more than one circle that satisfies the conditio ns. Express the final equations in the form \(x^{2}+y^{2}+D x+E y+F=0\). $$ \text { Center at }(-3,4) \text { and } r=2 \quad x^{2}+y^{2}+6 x-8 y+21=0 $$

Step-by-Step Solution

Answer

The equation is \(x^2 + y^2 + 6x - 8y + 21 = 0\).

Step 1: Identify the Center and Radius

The problem provides the center of the circle as \((-3, 4)\) and the radius as \(r = 2\). A circle's equation is typically given in the format \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Write the Equation in Standard Form

Using the center \((-3, 4)\) and radius \(r = 2\), substitute into the standard equation: \((x + 3)^2 + (y - 4)^2 = 4\).

Step 3: Expand the Equation to a General Form

Expand \((x + 3)^2 + (y - 4)^2 = 4\):\((x + 3)^2 = x^2 + 6x + 9\)\((y - 4)^2 = y^2 - 8y + 16\)Combine these:\(x^2 + 6x + 9 + y^2 - 8y + 16 = 4\).

Step 4: Simplify the General Form of the Equation

Combine like terms on the left side: \(x^2 + y^2 + 6x - 8y + 25 = 4\).Subtract 4 from both sides to get it into the required form:\(x^2 + y^2 + 6x - 8y + 25 - 4 = 0\).This simplifies to:\(x^2 + y^2 + 6x - 8y + 21 = 0\).

Key Concepts

Circle Center and Radius

The center and radius of a circle are fundamental in defining the circle's shape and position in a two-dimensional plane. The center, represented as \(h, k\), is the point from which all points on the circle are equidistant. The distance from this center to any point on the circle is known as the radius, represented by \(r\).

In the exercise, the circle's center is given as \((-3, 4)\), and the radius is \(2\). This means that every point on this circle is exactly \(2\) units away from the point \((-3, 4)\). Understanding these parameters is crucial as they directly feed into crafting the circle's equation, which sets the groundwork for any further transformations or manipulations of the equation.

Standard Form of Circle Equation

The standard form of a circle's equation is a critical tool for identifying its geometry and simplifying its graphical representation. It is written as \((x-h)^2 + (y-k)^2 = r^2\), where \(h\) and \(k\) indicate the center's coordinates, and \(r\) symbolizes the radius.

In the given problem, the center \((-3, 4)\) and radius \(2\) are substituted directly into this equation. This results in \((x+3)^2 + (y-4)^2 = 2^2\). By employing this formula, students can easily visualize the circle and understand its foundational structure. Mastering the standard form is essential, as it allows for quick interpretation and further manipulation when transforming the equation into other forms.

Converting to General Form

Once you have the circle equation in its standard form, the next step is transforming it to its general form for further computational ease. This involves expanding the terms to simplify and merge them into a singular expression. The general form of a circle's equation is given as \(x^2 + y^2 + Dx + Ey + F = 0\).

To convert \((x+3)^2 + (y-4)^2 = 4\) into this form, expand the squared terms: \(x^2 + 6x + 9\) and \(y^2 - 8y + 16\).

Combining these into a single equation, \(x^2 + 6x + 9 + y^2 - 8y + 16 = 4\), sets the stage to solve for specific constants \(D, E, ext{ and } F\). This expands the simplicity of the circle's nature by transitioning from a visual-centric to a more algebraic perspective. Understanding the algorithm of expanding and regrouping these terms is invaluable for algebra-related tasks.

Simplifying Equations

Simplifying equations is about clearing complexities to present a clear and straightforward expression. For circle equations, simplification often means aligning it to the form \(x^2 + y^2 + Dx + Ey + F = 0\).

In the exercise, the next step after expansion is combining like terms and balancing both sides: \(x^2 + y^2 + 6x - 8y + 25 = 4\).

Then, subtract the \(4\) on the right to move all terms to one side, resulting in: \(x^2 + y^2 + 6x - 8y + 21 = 0\).

This refined equation language ensures that every term is directly related to understanding the circle's size and position. By sharpening equations, mathematicians can make cleaner inferences and, importantly, accurate computations. Simplifying plays a foundational role in transitioning from drawable diagrams to computational analysis.