Problem 15 - Algebra 2

Question

Identify the conic section represented by each equation by writing the equation in standard form. For a parabola, give the vertex. For a circle, give the center and the radius. For an ellipse or a hyperbola, give the center and the foci. Sketch the graph. \(x^{2}+y^{2}-2 x+6 y=3\)

Step-by-Step Solution

Answer

The conic section represented by the equation is a circle. The standard form of the equation is \( (x - 1)^2 + (y + 3)^2 = 13 \). The center of the circle is at (1, -3) and the radius is \( \sqrt{13} \).

Step 1: Rearrange the equation

First, rearrange the equation and group the x and y terms. So, the equation becomes: \( (x^{2}-2x) + (y^{2} + 6y) = 3 \)

Step 2: Complete the square for x and y terms

To complete the squares: for the x-terms, take half of the coefficient of x, square it and add it to both sides. This gives: \( (x^{2} - 2x + 1) = 1 \). For the y-terms, take half of the coefficient of y, square it and add it to both sides. This gives: \( (y^{2} + 6y + 9) = 9 \). Add these equations together to give: \( (x^{2} - 2x + 1) + (y^{2} + 6y + 9) = 3 + 1 + 9 = 13 \)

Step 3: Convert into standard form of a circle

Now the equation looks like this: \( (x - 1)^2 + (y + 3)^2 = 13 \). This is the standard form of a circle with center at (h, k) and radius r, where: \( (x - h)^2 + (y - k)^2 = r^2 \). Compare this to the standard form to find the center and radius of the circle.

Step 4: Identify the center and radius of the circle

From the standard form of the equation of the circle, we can see that the center of the circle is at (1, -3), and 13 is equal to the radius squared, so the radius of the circle is: \( \sqrt{13} \).

Step 5: Sketch the graph

A sketch of the graph would have the center of the circle at (1, -3), and the radius of the circle would be \( \sqrt{13} \) units out from the center in all directions.

Key Concepts

Standard Form of a Circle

In geometry, the standard form of a circle is an essential concept that helps in identifying and solving various geometric problems. The standard form of a circle equation is structured as follows: \[(x - h)^2 + (y - k)^2 = r^2\]In this equation,

• \( (h, k) \) is the center of the circle.
• \( r \) is the radius of the circle.

Writing the equation in this form makes it very straightforward to identify both the center and the radius of the circle, key elements for graphing and analysis. By rewriting any generic equation of a circle to its standard form, we simplify the problem and make it easier to interpret.

Completing the Square

Completing the square is a useful algebraic technique for solving equations and rewriting expressions in a more convenient form. This method is particularly useful when transforming a quadratic expression into a perfect square trinomial.
To complete the square, follow these steps:

• Identify the quadratic and linear terms you wish to complete. For example: \( x^2 - 2x \)
• Take half of the coefficient of the linear term (\( -2 \) in this case), square it, and add it to both sides of the equation. This yields: \( (x^2 - 2x + 1) \)
• Repeat the process for the y terms: \( y^2 + 6y \) becomes \( (y^2 + 6y + 9) \)

By completing the square, you can transform any equation into an equation of a circle in standard form, making it easier to identify characteristics like center and radius.

Equation of a Circle

The equation of a circle in mathematics is pivotal when dealing with various geometric problems. It reflects the set of all points in a plane that are a fixed distance, called the radius, from a given point, called the center.

The equation is usually expressed in its standard form, described earlier, for clarity and ease of interpretation. Solving problems involving circles typically involves converting given equations into the standard circle form by using algebraic techniques such as completing the square.

Once in standard form, reading off the properties of the circle—its center and radius—becomes straightforward, allowing for easier graphing and geometric calculations.

Circle Center and Radius

The center and the radius are fundamental properties of a circle. Identifying these from an equation allows us to fully describe and graph the circle.
The center, denoted as \( (h, k) \) in the standard form of the equation, is the point equidistant from all points on the circle. The radius \( r \) of the circle is the distance from the center to any point on the circle. This relationship is clear in the standard form: \[(x - h)^2 + (y - k)^2 = r^2\]To find the center and radius from an equation not originally in standard form, one must manipulate the equation—often using techniques like completing the square. In our example, converting the equation to \[(x - 1)^2 + (y + 3)^2 = 13\],we see that:

• Center: \( (1, -3) \)
• Radius: \( \sqrt{13} \)

Accurately calculating the center and radius of a circle ensures precise graphing and understanding of circle-related problems.