Problem 9 - Algebra 2

Question

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ 6 x^{2}+6 y^{2}=162 $$

Step-by-Step Solution

Answer

The equation is a circle with radius \(3\sqrt{3}\) centered at (0,0).

Step 1: Write the Equation in Standard Form

The given equation is \(6x^2 + 6y^2 = 162\). To write this equation in standard form, which is typically \(Ax^2 + By^2 + Cx + Dy + E = 0\) for conic sections, we need to simplify it first. Divide every term by 6 to find:\[x^2 + y^2 = 27\]This equation is already simplified and in the standard form for a circle.

Step 2: Identify the Type of Conic

The equation \(x^2 + y^2 = 27\) indicates a circle because it is of the form \(x^2 + y^2 = r^2\), where \(r\) is the radius. Here, \(r^2 = 27\), so the conic is a circle.

Step 3: Determine the Radius and Center

For the equation \(x^2 + y^2 = 27\), the center of the circle is at \((0,0)\) since there are no additional terms shifting the circle's center on the x or y axis. The radius \(r\) can be determined by taking the square root of 27:\[r = \sqrt{27} = 3\sqrt{3}\approx 5.2\]

Step 4: Graph the Circle

To graph the circle, plot the center of the circle at \((0,0)\). Using the radius \(3\sqrt{3}\) or approximately 5.2, draw a circle centered at \((0,0)\) with this radius. It will extend from roughly \(-5.2\) to \(5.2\) along both the x and y axes.

Key Concepts

Standard Form

The standard form of a conic section helps us to easily identify the type of conic and its properties just by looking at the equation. For circles, the equation generally appears in the form \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle.
In our exercise, the equation given is \(6x^2 + 6y^2 = 162\). To convert this into the standard form, we simplify by dividing the entire equation by 6, resulting in \(x^2 + y^2 = 27\).
This final equation indicates that it is a circle with no linear terms, thus confirming its pure circular shape centered at the origin. Identifying the standard form is an important step, as it lays the foundation for understanding and graphing conics correctly.

Circle Equation

The circle equation in its standard form is crucial for graphing and understanding circles. For the given equation \(x^2 + y^2 = 27\), it represents a circle. This form fits into the general circle equation \(x^2 + y^2 = r^2\). The lack of \(x\) or \(y\) terms indicates the center of the circle is at the origin \((0,0)\).
The term \(r^2 = 27\) tells us the circle's radius can be found by taking the square root. Calculating \(r = \sqrt{27} = 3\sqrt{3}\), provides the actual size of the circle. This radius translates to approximately 5.2. But keeping it in the exact form \(3\sqrt{3}\) is more accurate for mathematical calculations.
Thus, the circle's properties, such as center and radius, are easily discerned from the equation in its standard form.

Graphing Circles

Graphing circles becomes straightforward once we know the circle equation and its properties. Start by plotting the center of the circle, which for \(x^2 + y^2 = 27\) is \((0,0)\). This central point is crucial, as all other points on the circle will be equidistant from it, at a distance of the radius.
Next, since the radius is \(3\sqrt{3}\) or approximately 5.2, you can measure exactly this distance from the center in all directions to plot the circle's boundary. In practical terms, you'll mark points approximately 5.2 units away vertically, horizontally, and diagonally, ensuring smooth rounding between them.
Using these points, you can draw the circle freehand or with the aid of a compass for precision. This visualizes how the equation lays out in the coordinate plane, perfectly illustrating the circle's symmetry and uniformity.