Problem 16 - Algebra 2

Question

Write an equation for each translation. $$ x^{2}+y^{2}=50 ; \text { right } 5 $$

Step-by-Step Solution

Answer

The equation of the circle after the translation to the right by 5 units is \((x-5)^{2}+y^{2}=50\).

Step 1: Determine the original center of the circle

The given equation is of the form \(x^{2}+y^{2}=r^{2}\) where (h, k) is the center and r is the radius of the circle. The coordinates of the center of the circle are given by (-h, -k). In our equation, h and k are both 0. Thus, the center of our circle is at the origin (0,0).

Step 2: Translate the circle

A translation to the right by 5 units will change the x-coordinate of the center from 0 to 5. We don't change the y-coordinate as the circle hasn't moved up or down. Hence, the center of the translated circle is now (5,0).

Step 3: Formulate the equation of the translated circle

The equation of a circle in the translated form is \((x-h)^{2}+(y-k)^{2}=r^{2}\), substituting \(h = 5\), \(k = 0\) and \(r = \sqrt{50}\), we get \((x-5)^{2}+y^{2}=50\). This equation represents the circle after it has been translated 5 units to the right.

Key Concepts

Equation of a Circle

An equation of a circle is a way to describe the circle's position and size on the coordinate plane. When we hear the term 'equation of a circle,' we usually refer to its standard form: \((x - h)^2 + (y - k)^2 = r^2\). Here, \(h\) and \(k\) represent the coordinates of the circle's center, and \(r\) is the radius.

To understand this equation, consider its components:

• \((x - h)^2\): This shifts the circle \(h\) units horizontally. If \(h\) is positive, it moves right; if negative, it moves left.
• \((y - k)^2\): This shifts the circle \(k\) units vertically. Positive means up; negative means down.
• \(r^2\): This defines the size of the circle. The larger the \(r\), the bigger the circle.

One of the simplest forms of a circle's equation is when the center is the origin \((0, 0)\), which can be written as \(x^2 + y^2 = r^2\).

This gives us a starting point for translating and transforming circles on the coordinate plane.

Coordinate Geometry

Coordinate geometry, or analytic geometry, uses a coordinate system to explore geometric problems. It bridges algebra with geometry, making it possible to solve geometric problems algebraically.

This approach is very powerful, especially when we talk about transformations such as translations. Let's break down some core aspects:

• Origin: The point \((0, 0)\) on a plane, where the x-axis and y-axis intersect. Many geometric figures are considered from this reference point.
• Points and Distances: Points on this plane are defined by \((x, y)\) coordinates. The distance between points is computed using the distance formula based on the Pythagorean theorem.
• Transformations: These include shifts, rotations, and reflections, which can be expressed algebraically and graphically.

One powerful transformation is the translation, which moves every point of a figure the same distance in a given direction. In the exercise above, the circle's center is translated 5 units to the right, demonstrating a common application of coordinate geometry.

Algebraic Manipulations

Algebraic manipulations involve transforming an equation or expression to simplify or solve it. These transformations make it easier to work with mathematical expressions, and understanding them is crucial when dealing with the equations of geometric shapes like circles.

Consider our transformation of the circle's equation during its translation:

• Substitution: Once we identify the new center \((h, k)\) after translation, we substitute these values into the general form \((x-h)^2 + (y-k)^2 = r^2\).
• Simplification: Through manipulation, we may simplify or expand equations to better interpret them or compare to other forms.
• Understanding Constants: Recognizing constants such as the radius \(r\) helps in maintaining the properties of the circle during manipulations.

In the provided exercise, substituting \(h = 5\), \(k = 0\) directly into the equation and acknowledging the radius as \( \sqrt{50} \) are essential steps.

These manipulations allow us to see the transformation from the original circle to its new position in a visual and calculative sense.