Problem 25 - 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. \(x^{2}+y^{2}=5 ;\) left 5 units, upward 3 units

Step-by-Step Solution

Answer

Translated circle equation: \((x + 5)^2 + (y - 3)^2 = 5\), center: \((-5, 3)\), radius: \(\sqrt{5}\).

Step 1: Identify Original Circle Equation

The given equation is \(x^2 + y^2 = 5\). This is the equation of a circle centered at the origin \((0, 0)\) with a radius of \(\sqrt{5}\).

Step 2: Translate the Circle Left by 5 Units

To move the circle to the left by 5 units, we need to add 5 to the \(x\)-coordinate of the center. This changes the \(x\) part of the equation from \(x^2\) to \((x + 5)^2\).

Step 3: Translate the Circle Upward by 3 Units

To move the circle upward by 3 units, we need to subtract 3 from the \(y\)-coordinate of the center. This changes the \(y\) part of the equation from \(y^2\) to \((y - 3)^2\).

Step 4: Write the Equation for the Translated Circle

Substituting the changes from Steps 2 and 3, the translated circle's equation is \((x + 5)^2 + (y - 3)^2 = 5\).

Step 5: State the New Center and Radius

The translated circle's center is \((-5, 3)\) and its radius remains \(\sqrt{5}\), as the radius does not change during a translation.

Key Concepts

Geometric Transformations

When it comes to working with circles and other shapes in geometry, one of the key concepts to understand is geometric transformations. These transformations allow us to change the position, size, and orientation of a shape.
They consist mainly of translations (sliding), rotations (turning), reflections (flipping), and dilations (resizing).
This exercise focuses on translation, where we slide the entire shape without altering its size or orientation.

• Translations: Moving a shape up, down, left, or right.
• Rotations: Spinning the shape around a fixed point.
• Reflections: Creating a mirror image over a line.
• Dilations: Growing or shrinking a shape while keeping proportions.

Understanding these transformations is essential because they help maintain the key properties of shapes while moving or resizing them in a coordinate plane.

Translating Shapes

Translating shapes is all about moving them from one location to another in a plane. Unlike other transformations, translation keeps the shape’s size, angle, and orientation the same.
In this process, each point of the shape moves in the same direction and by the same distance. For a circle, which is determined by its center point and radius, this simply involves shifting the center.
To translate a shape:

• Identify the direction of the translation (left, right, up, down).
• Adjust the coordinates of the shape's defining points accordingly.

In our specific example, the circle is moved "left by 5 units" and "up by 3 units." This results in a new equation that reflects this shift in the center coordinates.

Center of a Circle

The center of a circle is a point from which all points on the circle are equidistant. It is one of the defining elements of a circle in the coordinate plane and is usually denoted as \(h, k\).
This center can be easily identified from the equation of a circle, which generally takes the form \( (x - h)^2 + (y - k)^2 = r^2 \).

Impacts of Shifting the Center:

• Shifting the center affects only the \(h, k\) values in the circle's equation.
• The new center represents the translated position in the plane.

In our exercise, the circle's initial center is at \(0, 0\). After translating it left by 5 units and up by 3 units, the center becomes \(-5, 3\). This shift is reflected by changing the equation to \( (x + 5)^2 + (y - 3)^2 = 5 \).

Radius of a Circle

The radius of a circle is the distance from the circle's center to any point on its circumference. It is a crucial part of the circle's equation, reflected in the term \( r^2 \).

Key Features of Radius:

• Remains constant during any translation of the circle.
• Directly determines the size and area of the circle.

In the original equation \(x^2 + y^2 = 5\), the radius is \(\sqrt{5}\) since the right side of the equation represents \(r^2\).
Even when the circle is translated or its center is shifted, the radius does not change. That means after the transformation, the circle still has the same radius of \(\sqrt{5}\), keeping the circle’s size consistent.