Problem 18 - 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. \(x^{2}+y^{2}+14 y=-13\)

Step-by-Step Solution

Answer

The given equation represents a circle with the center at (0, -7) and a radius of 6.

Step 1: Rearrange the equation into a standard form

Given the equation \(x^{2}+y^{2}+14 y=-13\), we can rearrange it. The equation can be rewritten as \(x^{2}+(y+7)^{2}=36\) by completing the square on the y terms. Now, this is the standard form equation of a circle, which is \((x-a)^{2}+(y-b)^{2}=r^{2}\).

Step 2: Identify the conic section and its properties

In the standard form \((x-a)^{2}+(y-b)^{2}=r^{2}\), 'a' and 'b' are the x and y coordinates of the center of the circle and 'r' is the radius. We can see that in our equation \(x^{2}+(y+7)^{2}=36\), the center of the circle is at (0, -7) and the radius is \(\sqrt{36}\), which equals 6.

Step 3: Sketch the graph

To draw the graph, first mark the center of the circle at (0, -7) on the graph. Then draw a circle with a radius of 6 units. The circle will touch the y-axis at points (0, -1) and (0, -13).

Key Concepts

Standard Form Equation

The standard form equation of a conic section is essential in identifying the particular type of conic and its key properties. For circles, the standard form is written as: \[(x-a)^{2} + (y-b)^{2} = r^{2}\] where

• \(a\) and \(b\) are the x and y coordinates of the circle's center
• \(r\) is the radius of the circle

This setup allows us to quickly recognize and analyze circles by visualizing their geometric properties such as the center and size. By converting given equations into this format, identifying circle properties becomes straightforward. To get an equation into standard form, rearrange it by isolating and completing the square for each variable. This process might seem complex, but with consistent practice, it becomes a systematic approach to understanding conics in standard form.

Circle Properties

Circles have unique characteristics that differentiate them from other conic sections. Understanding these properties can help you recognize and work with circles more effectively. Key properties include:

• **Center**: The fixed point around which the circle is drawn. In a standard form equation like \((x-a)^{2} + (y-b)^{2} = r^{2}\), the center is \((a, b)\).


• **Radius**: The distance from the center to any point on the circle. It is given by \(r\) in the equation, which is the square root of the value on the right-hand side.

To extract these properties from an equation, identify the standard form first. From there, recognizing the center as well as the radius becomes a simple step. These properties are essential when sketching or analyzing circles mathematically and geometrically.

Completing the Square

Completing the square is a method used to transform quadratic expressions into a form that makes solving equations simpler. It is particularly useful when dealing with conic sections like circles. ### How to Complete the Square:1. **Identify the Quadratic Expression**: Look for expressions of the form \(ax^2 + bx\).2. **Half the Coefficient of x**: Take \(b\), halve it and square the result. This is called the "square term."3. **Adjust the Equation**: Add and subtract the "square term" inside the equation to help reorganize it into a perfect square trinomial.For example, in the equation \(x^{2} + y^{2} + 14y = -13\), focusing on \(y\):

• The constant term with \(y\) is 14. Half of it is 7, and 7 squared is 49.
• Reorganize. Add 49 inside the equation where appropriate to complete the square: \(x^{2} + (y + 7)^{2} = 36\).

Completing the square takes practice but is crucial for manipulating equations into forms that reveal properties of conic sections, especially circles with clarity.