Problem 17 - 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. $$ x^{2}+y^{2}+6 y+13=40 $$

Step-by-Step Solution

Answer

The equation represents a circle and its graph is centered at (0, -3) with radius 6.

Step 1: Convert to Standard Form

Start by rewriting the given equation: \( x^2 + y^2 + 6y + 13 = 40 \). First, move the constant term to the right side: \( x^2 + y^2 + 6y = 27 \). Next, complete the square for the \(y\) terms.

Step 2: Complete the Square

Take the equation \( x^2 + y^2 + 6y = 27 \). To complete the square on \( y \), use the formula \((b/2)^2\) where \( b \) is the coefficient of \( y \). Here, \( b = 6 \). So, \((6/2)^2 = 9\). Add and subtract 9 inside the equation: \( x^2 + (y^2 + 6y + 9 - 9) = 27 \).This becomes: \( x^2 + (y + 3)^2 - 9 = 27 \).Simplify to: \( x^2 + (y + 3)^2 = 36 \).

Step 3: Identify the Conic Section

The equation \( x^2 + (y + 3)^2 = 36 \) is in the form \((x-h)^2 + (y-k)^2 = r^2\), which is the standard form for a circle.Here, the center is \((h, k) = (0, -3)\) and the radius \( r = 6 \). So, the graph is a circle.

Key Concepts

circle

A circle is a perfect geometrical shape where every point on the edge is equidistant from a central point, known as the center of the circle. This distance from the center to any point on the circle is called the radius.
In mathematics, circles are one of the basic shapes studied in the branch of geometry called conic sections.

• A circle can be derived from slicing a cone horizontally parallel to its base.
• Its characteristics include uniform symmetry, constant curvature, and a simple central point forming symmetry around it.
• It is often represented in an equation form which helps in graphical analysis.

Understanding the nature of a circle is crucial because it forms the basis for learning about other conic sections like ellipses, parabolas, and hyperbolas, which have a variety of applications in fields ranging from engineering to astronomy.

standard form

The standard form of a circle is a way to express its equation in the most compact and recognizable manner. This form sets the foundation for easily interpreting and graphing the circle.

• The general representation is: \((x-h)^2 + (y-k)^2 = r^2\).
• Here, \((h, k)\) is the center of the circle, and \(r\) is the radius.

For the circle's equation, relocating it from its general equation to the standard form allows us to quickly identify the circle's core features:

• Its center
• Its radius.

By transforming to this form, analysis and graph interpretation become much easier, thereby simplifying many practical applications.
Learners can apply this knowledge to solve and graph equations efficiently in various mathematical and real-world problems.

complete the square

Completing the square is a key algebraic technique used to manipulate a quadratic equation into a form that reveals significant properties about the shape of its graph. This method is especially helpful in converting an equation into the standard form.
When applied to equations of conic sections:

• This method helps specify the center and radius of circles or vertex of parabolas, making it easier to graph them.
• In our example, we had to complete the square for the \(y\) terms in the equation \(x^2 + y^2 + 6y = 27\).
• First, identify the coefficient \(b\) in the \(y\) terms, and use the formula \((b/2)^2\) to find the value to add and subtract for completing the square.
• This gives us the expression \((y + 3)^2\), turning the original equation into a recognizable, standard form.

Mastering this technique enhances problem-solving skills and allows for deeper understanding in graphing and analyzing conic sections.