Problem 54 - Precalculus

Question

Find the standard equation of the circle tangent to the \(x\) -axis and with center \((3,5) .\) Hint: First draw a sketch.

Step-by-Step Solution

Answer

The standard equation of the circle is \((x-3)^2 + (y-5)^2 = 25\).

Step 1: Understand the Problem

We need to find the equation of a circle that is tangent to the x-axis with the center at (3,5). Since the circle is tangent to the x-axis, the distance from the center to the x-axis must equal the radius of the circle.

Step 2: Determine the Radius

The center of the circle is at (3,5). Since the circle is tangent to the x-axis, the distance from the center to the x-axis (y = 0) is the y-coordinate of the center, which is 5. Therefore, the radius of the circle is 5.

Step 3: Write the Standard Equation of the Circle

The standard equation of a circle with center \((h, k)\) and radius \(r\) is \((x-h)^2 + (y-k)^2 = r^2\). For this circle, the center is at \((3,5)\) and the radius is \(5\). Substitute these values into the formula to get: \((x-3)^2 + (y-5)^2 = 5^2\).

Step 4: Simplify the Equation

Simplify the equation to express it in the standard form. \(5^2 = 25\), so the equation becomes \((x-3)^2 + (y-5)^2 = 25\).

Key Concepts

Center of a Circle

To find the center of a circle, you need to identify the point that is equidistant from all points on the circle. In the context of a geometric plane, the center is represented as a coordinate pair

• The horizontal component, denoted by the letter \(h\),
• And the vertical component, denoted by the letter \(k\).

The center of a circle gives you a reference point from which various elements of circle-related calculations can be determined. In the given example, the circle's center is at the point

• \((3,5)\), meaning that 3 is the horizontal distance from the vertical axis,
• and 5 is the vertical distance from the horizontal axis.

This center point is crucial because it helps define the equation of the circle in its standard form.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. It is a fundamental circle component because it directly influences the size of the circle. In mathematical terms, the radius is represented by the letter \(r\). To find the radius in a coordinate plane, you usually measure the distance from the circle's center to a point along the circle's boundary.

In the given exercise, the circle is tangent to the x-axis, meaning it just touches the x-axis without crossing it. This indicates that the radius is the vertical distance between the center of the circle and the tangent point on the x-axis. Since our center is at

• \((3,5)\),
• the radius is 5, the same as the vertical component of the center.

Understanding the radius allows us to complete the equation of the circle.

Tangent to the x-axis

When a circle is tangent to the x-axis, it means that the circle meets the axis at exactly one point. The term "tangent" describes how a line just touches a curve at a single point without intersecting it.

For a circle tangential to a line such as the x-axis, the radius is perpendicular to this tangent point. Therefore, when you know the circle's center, the tangent property is useful for determining the radius. In our example:

• The y-coordinate of the circle's center is 5, indicating the exact distance from the center to the x-axis.
• Thus, the circle is tangent to the x-axis at the point where the radius equals the y-coordinate.

This tangency provides you a straightforward path to understanding circle behavior relative to a line and assists in deriving the complete circle equation.