Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(11-16\), find the equation of the circle satisfying the given conditions. Center \((3,4)\) and tangent to \(x\)-axis

Step-by-Step Solution

Answer

The equation is \((x - 3)^2 + (y - 4)^2 = 16\).

Step 1: Understand the Problem

The problem gives the center of a circle at the point \((3, 4)\) and states that the circle is tangent to the \(x\)-axis. "Tangent to the \(x\)-axis" means the circle just touches the \(x\)-axis at one point.

Step 2: Identify the Radius

Since the circle is tangent to the \(x\)-axis, the distance from the center of the circle to the \(x\)-axis will be equal to the radius of the circle. The center is at \((3, 4)\), so the y-coordinate \(4\) is the distance from the center to the \(x\)-axis. Hence, the radius \(r = 4\).

Step 3: Use the Circle Equation Formula

A circle with center \((h, k)\) and radius \(r\) can be described by the equation: \((x - h)^2 + (y - k)^2 = r^2\). Here, \(h = 3\), \(k = 4\), and \(r = 4\).

Step 4: Substitute Values into the Equation

Substitute the values of \(h\), \(k\), and \(r\) into the circle equation: \((x - 3)^2 + (y - 4)^2 = 4^2\). Calculate the radius squared: \(4^2 = 16\).

Step 5: Write the Final Equation

The equation of the circle is: \((x - 3)^2 + (y - 4)^2 = 16\).

Key Concepts

circle geometry

The realm of circle geometry is fascinating and essential to many mathematical pursuits. The most fundamental element of a circle is its equation, which embodies the set of all points equidistant from a single point, known as the center. Understanding this equation helps unlock the circle's mystery.

The standard equation of a circle with its center at \(h, k\) and radius \(r\) is given by: \\[ (x - h)^2 + (y - k)^2 = r^2 \]Here, we've identified each part:

• \( (x - h) \) Indicates the horizontal distance.
• \( (y - k) \) Refers to the vertical distance.
• \( r^2 \) Represents the square of the radius.

With the circle's center at \(3,4\), our problem situates us in a geometric space defined by these principles, allowing us to comprehend the circle's reach in terms of both axis coordinates.

Having a circle centered at a point with a specific tangent informs these geometrical properties to provide insight into how the circle touches external lines or coordinates.

tangent line

Grasping the concept of a tangent line is pivotal in circle geometry. A tangent line to a circle is a straight line that touches the circle at exactly one point. This single point is crucial and defining, as it means the circle and the line do not intersect at any other point.

For a circle centered at \(3, 4\) and tangent to the \(x\)-axis:

• The tangent line here is horizontal, running along the \(x\)-axis.
• The contact point happens when the \(y\)-coordinate equals zero, making it just touch the circle.

Knowing that the circle is tangent to the \(x\)-axis signifies that the radius stretches vertically downward to this tangent point, validating the radius as the \(y\)-coordinate of the center itself.

radius calculation

In the study of circles, calculating the radius is a direct pathway to defining the circle's overall size. The radius is the constant distance from the center of the circle to any point on its perimeter.

Given a circle centered at the point \( (3, 4)\) and tangent to the \(x\)-axis, how do we ascertain the radius? The term "tangent to the \(x\)-axis" immediately hints at a vertical stretch from the center. Here’s how:

• The distance from the center to the \(x\)-axis, which is the "tangency point," equals the \(y\)-coordinate of the circle's center.
• This \(y\)-coordinate is \(4\), thus making the radius \(4\).

Therefore, \(r = 4\). It's these seemingly simple measurements that craft the shape and scope of the circle, serving practical purposes in both theory and real-world applications.