Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(11-16\), find the equation of the circle satisfying the given conditions. Center \((4,3)\), goes through \((6,2)\)

Step-by-Step Solution

Answer

The circle's equation is \((x - 4)^2 + (y - 3)^2 = 5\).

Step 1: Understand the Circle Equation

The general equation of a circle in the coordinate plane is given by \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius. We need to find the equation of the circle with center \((4, 3)\) going through the point \((6, 2)\).

Step 2: Identify the Circle's Parameters

We are given the center \((h, k) = (4, 3)\) of the circle. The next step is to determine the radius \(r\).

Step 3: Calculate the Radius

To find the radius \(r\), calculate the distance between the center \((4, 3)\) and the point \((6, 2)\), which lies on the circle. The distance formula is \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 4: Plug Points into Distance Formula

Using the distance formula, we calculate: \[r = \sqrt{(6 - 4)^2 + (2 - 3)^2} = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}\]. So, the radius \(r = \sqrt{5}\).

Step 5: Formulate the Circle's Equation

Plug the center \((4, 3)\) and \(r = \sqrt{5}\) into the circle equation, yielding: \((x - 4)^2 + (y - 3)^2 = 5\).

Key Concepts

Radius Calculation

In geometry, particularly when dealing with circles on the coordinate plane, calculating the radius is a vital step. The radius of a circle is the constant distance between its center and any point lying on its circumference.
To compute this distance, we utilize the well-known distance formula, which effectively measures the separation between two points. In this context, we're determining the radius length from a known center to a point the circle passes through. Ensure that you carefully identify both points (center and circumferential point), as any miscalculation here can lead to errors in further steps.

Distance Formula

The distance formula is a fundamental tool in geometry, allowing us to find the distance between any two points on a coordinate plane. The formula is derived from the Pythagorean Theorem and is written as:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of the two points.
• The terms \((x_2 - x_1)^2\) and \((y_2 - y_1)^2\) represent the square of the differences in the x-coordinates and y-coordinates respectively.

Simply plug in your point values to find the precise distance — in the context of our problem, this provides the radius of the circle.

Geometry

Geometry, particularly the study of circles, is foundational in mathematics and helps us understand shape, size, and the relative positions of figures. Circles have unique properties:

• They are completely symmetrical.
• Every point on the circle's edge is equidistant from the center.
• The circle's equation \((x - h)^2 + (y - k)^2 = r^2\) elegantly ties these properties together through its center \((h, k)\) and radius \(r\).

Approaching circle problems geometrically allows you to visually represent and solve problems, making abstract concepts more concrete and relatable.

Coordinate Plane

The coordinate plane is a two-dimensional surface where points are defined by their distance from two intersecting lines: the x-axis and the y-axis. Every point is represented by an ordered pair \((x, y)\):

• The x-coordinate indicates how far along the point is horizontally from the y-axis.
• The y-coordinate shows the vertical distance from the x-axis.

In problems involving circles on a coordinate plane, plotting the center and the circumference point gives a visual representation of the circle. Proper understanding of the coordinate plane simplifies the abstraction, helping solve equations and understand the placement and sizing of geometric figures such as circles.