Problem 75 - Algebra and Trigonometry

Question

\(73-76\) Sketch the graph of the equation. $$ x^{2}+y^{2}+6 x-12 y+45=0 $$

Step-by-Step Solution

Answer

The graph is a point at (-3, 6).

Step 1: Rearrange the Equation

First, rewrite the given equation. We have: \[ x^2 + y^2 + 6x - 12y + 45 = 0 \] Rearrange it to group the \(x\) and \(y\) terms separately as follows: \[ (x^2 + 6x) + (y^2 - 12y) = -45 \]

Step 2: Complete the Square for x

Complete the square for the \(x\)-terms in the equation. Take the coefficient of \(x\), which is 6, divide it by 2 to get 3, and then square 3 to get 9. Add and subtract 9 inside the \(x\)-bracket:\[ (x^2 + 6x + 9) - 9 \] This becomes:\[ (x + 3)^2 - 9 \]

Step 3: Complete the Square for y

Complete the square for the \(y\)-terms. Take the coefficient of \(y\), which is -12, divide it by 2 to get -6, and square it to get 36. Add and subtract 36 inside the \(y\)-bracket:\[ (y^2 - 12y + 36) - 36 \]This becomes:\[ (y - 6)^2 - 36 \]

Step 4: Combine Completed Squares and Constants

Substitute the completed squares back into the equation and combine constants:\[ (x + 3)^2 - 9 + (y - 6)^2 - 36 = -45 \]Combining the constant terms gives:\[ (x + 3)^2 + (y - 6)^2 = 0 \]

Step 5: Interpret and Sketch the Graph

The equation \( (x + 3)^2 + (y - 6)^2 = 0 \) represents a degenerate circle which is a point at \( x = -3, y = 6 \). Sketch this as a single point on the coordinate plane.

Key Concepts

Degenerate Circle

In mathematics, a degenerate circle is a condition where what we would expect to be a circular shape actually reduces to a singular point. Normally, the equation of a circle is \[ (x - h)^2 + (y - k)^2 = r^2 \]where

• \( h \) and \( k \) are the coordinates of the circle's center
• \( r \) is the radius of the circle

However, if \( r \) equals zero, then the shape collapses into a point. This happens because every point on the circle's perimeter moves to the center, leaving us with just the center point. For the equation \[ (x + 3)^2 + (y - 6)^2 = 0 \]we identify the degenerate circle's center at point \( (-3, 6) \).Here, the radius \( r \) is \(0\), so it doesn't cover any area. Thus, degenerate circles like this represent singular points rather than extended shapes on the coordinate plane.

Equation Rearrangement

Equation rearrangement is a crucial step in simplifying and solving complex problems. It involves manipulating an equation to make it easier to work with, often by collecting like terms or organizing parts to aid further operations. For instance, in our exercise, the given equation \[ x^2 + y^2 + 6x - 12y + 45 = 0 \]is rearranged to\[ (x^2 + 6x) + (y^2 - 12y) = -45 \].This step organizes terms to allow us to complete the square efficiently for both \(x\) and \(y\) terms.
Rearrangement simply makes it clear what operations are needed next, paving the way for further factoring or simplification. Seeing these groups of terms separately helps highlight the paths or changes the equation will undergo, which is particularly useful when aiming to convert it into a more recognizable or standard form, like transforming it into a circle equation.

Graph Sketching

Graph sketching is a method used to visually represent mathematical equations on the coordinate plane. This visual representation helps in understanding the relations between different variables. For the equation converted into \[ (x + 3)^2 + (y - 6)^2 = 0 \],the graph is quickly drawn because it represents a degenerate circle, which is a single point rather than a typical circular shape.
The point (also the graph of the degenerate circle) is found at the coordinates \( (-3, 6) \).Graph sketching for a degenerate circle does not involve drawing a round shape but simply plotting a point where the circle's center lies. Recognizing such characteristics simplifies what needs to be visualized on a graph, differing from how regular shapes like circles and ellipses might be sketched.

Coordinate Plane

The coordinate plane serves as the background where equations come to life in graph form. It consists of two number lines perpendicular to each other: the horizontal \(x\)-axis and the vertical \(y\)-axis. Together, they allow us to plot points, lines, and shapes using pairs of numbers, known as coordinates. Each point is defined with an \(x\)-coordinate and a \(y\)-coordinate.
For example, the point in our exercise is \( (-3, 6) \).This tells us where to locate the degenerate circle on this plane. The beauty of working with the coordinate plane is that it offers a clear, visual interpretation of mathematical data, simplifying comprehension.

• The x-coordinate tells us how far to go left or right from the origin (0,0).
• The y-coordinate tells us how far to go up or down.

Understanding how to plot these coordinates is essential, as it forms the foundation for sketching all types of graph-based problems.