Problem 24 - College Algebra with Modeling and Visualization

Question

Write an equation that shifts the given circle in the specified manner. State the center and radius of the translated circle. \(x^{2}+y^{2}=9 ;\) right 2 units, downward 6 units

Step-by-Step Solution

Answer

Center: (2, -6); Radius: 3

Step 1: Identify the Original Equation

The original equation of the circle is given as \(x^2 + y^2 = 9\). This is in the standard form \((x-h)^2 + (y-k)^2 = r^2\) with center at \((0,0)\) and radius \(r = 3\).

Step 2: Apply Horizontal Translation

To shift the circle 2 units to the right, adjust the x-coordinate in the equation. Replace \(x\) with \((x-2)\): \((x-2)^2 + y^2 = 9\).

Step 3: Apply Vertical Translation

To shift the circle 6 units downward, adjust the y-coordinate. Replace \(y\) with \((y+6)\): \((x-2)^2 + (y+6)^2 = 9\).

Step 4: Identify the Translated Circle's Center and Radius

The new equation \((x-2)^2 + (y+6)^2 = 9\) shows that the translated circle's center is \((2, -6)\), and the radius remains \(3\).

Key Concepts

Equation of a Circle

An equation of a circle is a formula that helps define all the points within a plane that are equidistant from a fixed center point. The standard form of the circle equation is

• \((x-h)^2 + (y-k)^2 = r^2\)

Here,

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

In the provided example, the circle is initially defined by the equation \(x^2 + y^2 = 9\). This fits the standard form with

• a center at \((0,0)\),
• radius \(r = 3\) since \(r^2 = 9\).

When translating circles, these variables will change based on how much you move the circle across the coordinate plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe geometric figures through equations and numerics. This method helps us precisely translate and resize shapes within the coordinate plane.

• In terms of circle translation, coordinate geometry facilitates repositioning the circle by adjusting variables in the equation.
• Shifting right or left involves modifying the \(x\) value.
• Moving up or down changes the \(y\) value.

In our specific task, translating a circle to the right by 2 units means substituting \(x\) with \((x-2)\) in the equation. Translating downward by 6 units means substituting \(y\) with \((y+6)\). The equation will then reflect the circle's new position in the coordinate plane.

Geometric Transformations

Geometric transformations involve altering the position or size of a shape on a plane. The main types include translations, rotations, reflections, and dilations. Translations are the most fundamental since they involve moving a shape without altering its size or orientation.

• Translation keeps the shape's size and orientation consistent, only shifting its position to a different part of the plane.
• In the context of a circle, this means the radius remains the same while the central point \((h, k)\) shifts.

For example, to translate the circle \(x^2 + y^2 = 9\) to the right by 2 units and down by 6 units, the center relocates from \((0,0)\) to \((2,-6)\). The radius remains at 3, resulting in the translated equation \[(x-2)^2 + (y+6)^2 = 9\]This equation confirms that translations maintain the circle's size while changing its location on the graph.