Problem 29 - Algebra 2

Question

Write an equation for each conic section. Then sketch the graph. circle with center \((-6,9)\) and radius 9

Step-by-Step Solution

Answer

The equation of the circle is \(x^2 + y^2 + 12x - 18y + 45 = 0\) and it can be sketched by marking its center at (-6,9), then drawing the circle outline with radius 9 units.

Step 1: Write the equation of the circle

Substitute the given center coordinates into the standard form equation: \((x - (-6))^2 + (y - 9)^2 = 9^2\). Simplify to obtain the equation: \(x^2 + y^2 + 12x - 18y + 45 = 0\).

Step 2: Sketch the graph of the circle

On a graph paper, mark the center of the circle at point (-6,9). Draw a point at one radius distance away in each direction: up, down, left and right from the center. Then sketch the circle through these points to get a round shape.

Key Concepts

Circle Equation

The equation of a circle is a fundamental concept in the study of conic sections. To better understand it, imagine drawing a closed loop around a point on a plane, which we call the center. The equation defines all the points that form this loop at a consistent distance, known as the radius, from the center.
The standard form for the equation of a circle is \[(x - h)^2 + (y - k)^2 = r^2\],where:

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

In the original exercise, plugging in the values for the center \((-6, 9)\)and radius \(9\), the equation becomes:\[(x + 6)^2 + (y - 9)^2 = 81.\]
Upon simplification, it transforms into:\[x^2 + y^2 + 12x - 18y + 45 = 0.\]By understanding these components, writing the circle's equation becomes straightforward.

Graphing Conic Sections

Graphing conic sections involves representing geometric figures—such as circles, ellipses, parabolas, and hyperbolas—on a coordinate plane. In this scenario, we focus on how to graph circles. Follow these simple steps to graph a circle accurately using its equation.
First, identify the circle's center from the equation. For \(x^2 + y^2 + 12x - 18y + 45 = 0\), the center is \((-6, 9)\). Next, the radius can be found from the circle's equation, originally written in the form \((x + 6)^2 + (y - 9)^2 = 81\), with \(r = 9\).
To graph:

• Place a point at \((-6, 9)\), indicating the center.
• Measure \(9\) units in all cardinal directions starting from the center.
• Connect these outer points with a smooth, round curve to complete the circle’s outline.

Marking these points clearly and drawing a symmetric round shape are key to accurately representing a circle on a graph.

Geometric Transformations

Geometric transformations refer to the ways in which we can manipulate and change figures or shapes within a plane. These include translations, rotations, reflections, and scaling, all useful for converting geometric figures from one form to another while preserving certain properties.
In the context of conic sections, transformations help relate figures like circles, ellipses, and hyperbolas. For instance, to transform the equation \((x + 6)^2 + (y - 9)^2 = 81\), translated from the origin, where \((h, k)\) is \((-6, 9)\).

• Transformation accounts for moving the circle left by 6 units and up by 9 units from the original circle centered at the origin.
• The radius remains unchanged during translation.

These transformations ensure the circle maintains its diameter and shape while its position changes on the plane. Understanding transformations like these is critical for predictive and analytic approaches in geometry.