Problem 18 - A Graphical Approach to College Algebra

Question

Find the center-radius form for each circle satisfying the given conditions. Center \((5,-1) ;\) tangent to the \(y\) -axis

Step-by-Step Solution

Answer

The circle's equation is \((x - 5)^2 + (y + 1)^2 = 25\).

Step 1: Understand the Problem

The problem gives us the center of a circle \((5, -1)\) and states that the circle is tangent to the \(y\)-axis. Our goal is to find the equation of this circle in center-radius form.

Step 2: Identify the Radius

A circle is tangent to a line if the distance from the center of the circle to the line is equal to the radius of the circle. Since the circle is tangent to the \(y\)-axis and the center is \((5, -1)\), the radius is simply the \(x\)-coordinate of the center, which is 5.

Step 3: Write the Equation in Center-Radius Form

The general formula for a circle in center-radius form is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. From Step 1 and Step 2, we have \((h, k) = (5, -1)\) and \(r = 5\). Substitute these values into the formula:\[(x - 5)^2 + (y + 1)^2 = 5^2\]

Step 4: Simplify the Equation

Simplify the equation from Step 3:\[(x - 5)^2 + (y + 1)^2 = 25\]

Key Concepts

Center-Radius Form

The center-radius form is a way to express the equation of a circle using its center and radius. It's a very neat and practical format because it clearly highlights the circle's critical features. You can find this form being modeled as \((x - h)^2 + (y - k)^2 = r^2\), where:

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

The variables \(x\) and \(y\) are placeholders that represent any point on the circle. Whenever you need to specify a circle precisely, the center-radius form offers both precision and clarity. You simply substitute \((h, k)\) with the specific coordinates of your circle's center and \(r\) with its radius value.

Geometry

In geometry, understanding the properties of shapes is essential. When it comes to circles, two crucial elements are the center and the points on the circumference. The center, often depicted as \((h, k)\), serves as a symmetrical point around which all other points of the circle equidistantly revolve. When writing equations, like the center-radius form, geometry concepts ensure that the structure and form accurately embody the physical shape.

• The distance from the center to any point on the circle, known as the radius, forms an invisible boundary that defines the circle's size.
• Geometric rules ensure symmetry and consistency in identifying likeness among circles.

The relationship between algebra and geometry shines brightly through the lens of circle equations, where coordinates transform abstract ideas into concrete circular boundaries.

Tangency

Tangency is a fascinating concept in geometry that describes when a line touches a curve at exactly one distinct point. For circles, this means that at the point of tangency, the line is directly perpendicular to a line drawn from the center to the touching point. In this case, the given conditions state that the circle is tangent to the \(y\)-axis.

• This makes the distance from the center of the circle to the \(y\)-axis exactly equal to the circle's radius.
• Essentially, the circle just "kisses" the \(y\)-axis, perfectly aligning without crossing it.

Understanding tangency not only helps in visualizing the circle's size and position but also provides an elegant method of determining the radius when circle equations are involved.

Radius Calculation

Knowing how to calculate the radius is fundamental when working with circle equations. The radius is the distance between the circle's center and any point on the circumference. From the exercise, we were informed that the circle is tangent to the \(y\)-axis, which implies a strategic insight into finding the radius.

• Since the \(y\)-axis is vertical, the horizontal distance from \((5, -1)\) to the \(y\)-axis is simply the \(x\)-coordinate of the center, which is 5.
• Thus, without complex calculations, we can easily determine the radius as 5.

By deducing from the conditions provided, such as tangency and known center coordinates, radius calculations become both intuitive and straightforward in problems like these.