Problem 26 - 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. \(x^{2}+y^{2}=7 ;\) left 3 units, downward 7 units

Step-by-Step Solution

Answer

The translated circle's equation is \((x + 3)^2 + (y + 7)^2 = 7\). Center: \((-3, -7)\); Radius: \(\sqrt{7}\).

Step 1: Identify the original center and radius

For the circle equation in the standard form \(x^2 + y^2 = 7\), the center is at \((0, 0)\) and the radius is \(\sqrt{7}\).

Step 2: Determine the new center

Translate the center \((0, 0)\) by moving it left 3 units and downward 7 units. This shifts the center to \((-3, -7)\).

Step 3: Write the equation of the translated circle

The standard equation for a circle with center \((h, k)\) is \((x-h)^2 + (y-k)^2 = r^2\). Substituting \(h = -3\), \(k = -7\), and \(r^2 = 7\) gives the equation of the translated circle as \((x + 3)^2 + (y + 7)^2 = 7\).

Step 4: State the new center and radius

The new center of the translated circle is \((-3, -7)\), and the radius remains \(\sqrt{7}\).

Key Concepts

Translation of Circles

Translation of circles involves moving a circle from one position in the coordinate plane to another without altering its size or shape. It is essentially about sliding the circle along the x-axis and y-axis. This transformation is very helpful in geometry and can simplify many problems.

When you translate a circle, you'll need to carefully adjust its center's coordinates. For instance, if the original circle's center is at \(0, 0\) and you are instructed to move it left by 3 units, and down by 7 units, you simply apply these shifts:

• Moving left means subtracting from the x-coordinate: \(0 - 3 = -3\)
• Moving down means subtracting from the y-coordinate: \(0 - 7 = -7\)

This gives you a new center at \((-3, -7)\).

No matter how you translate the circle, the radius remains the same because translation doesn't affect size, only position. This makes translation a handy tool for visualizing and solving circle-related equations.

Center of a Circle

The center of a circle, denoted as \(h, k\), is a pivotal component in the equation of a circle. It defines the exact 'middle' of the circle in the coordinate plane. In the standard equation of a circle, \((x-h)^2 + (y-k)^2 = r^2\), \(h\) and \(k\) are the x and y coordinates of this center, respectively.

In practice, knowing the center allows you to understand the circle's position relative to the origin or other points on a plane. For example, in the exercise, the original circle had its center at \(0, 0\) because the equation was simply \(x^2 + y^2 = 7\). This form signifies that the circle is perfectly centered at the origin.

• When the center is moved to \((-3, -7)\), it shifts to the left and downward in the plane.
• This new positioning is vital for accurately graphing or analyzing the circle.

Understanding the center's role can significantly simplify many geometric and algebraic tasks, including sketching circles or converting between different forms of circle equations.

Radius of a Circle

The radius of a circle is a fixed distance from its center to any point on its circumference. In math, it's represented by \(r\) in the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\). The radius determines the size of the circle.

In cases where the equation is given in the form \(x^2 + y^2 = 7\), the radius is \(\sqrt{7}\), because \(r^2 = 7\). It is essential to extract the accurate radius from the equation to understand or recreate the circle's true proportions. Regardless of how far or in which direction the circle is translated, the radius remains unchanged. This aspect ensures consistency in the circle's dimension across various transformations.

• Knowing that the radius is \(\sqrt{7}\), whenever translated, helps keep the circle's size consistent.
• A stable radius implies that only position changes, not size or shape.

Understanding the radius's constancy across translations is fundamental for solving problems that involve moving circles in the coordinate plane.