Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the General Equation of a Circle

The general equation of a circle with center \((h, k)\) and radius \(r\) is given by: \((x - h)^2 + (y - k)^2 = r^2\). This formula represents the set of all points \((x, y)\) that are at a distance \(r\) from the center \((h, k)\).

Step 2: Identify Given Values for the Circle

We are given that the center of the circle is at \((-2, 3)\) and the radius \(r\) is 4. This means in the equation of the circle, \(h = -2\), \(k = 3\), and \(r = 4\).

Step 3: Substitute the Values into the Circle Equation

Substitute \(h = -2\), \(k = 3\), and \(r = 4\) into the general equation: \((x - (-2))^2 + (y - 3)^2 = 4^2\). This simplifies to \((x + 2)^2 + (y - 3)^2 = 16\).

Step 4: Write the Final Equation of the Circle

The equation of the circle with center \((-2, 3)\) and radius 4 is \( (x + 2)^2 + (y - 3)^2 = 16\).

Key Concepts

Center and Radius of a Circle

The center and radius are crucial when describing a circle on a plane. The center is a point represented by coordinates \(h, k\). These coordinates mark the exact middle point of the circle. The radius, on the other hand, is a fixed distance from this center to any point on the circle's edge.
For instance, if we have a circle with center \((-2, 3)\) and radius 4, it means every point on the edge of this circle is exactly 4 units away from \((-2, 3)\). This uniform distance defines the round shape of the circle.
When working with these values in math problems, it's important to clearly identify them from the given information. Often, they are directly provided or can be extracted from conditions that describe the circle.

General Equation of a Circle

Understanding the general equation of a circle is key to solving problems related to circles on a coordinate plane. This equation is written as \((x - h)^2 + (y - k)^2 = r^2\). Let's break down what each part means:

• \(x\) and \(y\) are variables representing any point on the circle.
• \(h\) and \(k\) are the coordinates of the circle's center.
• \(r\) is the radius of the circle, squared on the right-hand side of the equation.

This equation ensures that all points \(x, y\) stay at the radius's distance from the center \((h, k)\). This is how every possible point along the circle's circumference is defined mathematically.
By understanding this formula, you can easily find all the attributes of the circle and see how each part interacts to keep the circle's perfect shape intact.

Substitution in Circle Equation

Substitution is a vital step when working with circle equations. By placing known values into the general equation, you can find the specific equation for a given circle.
In our exercise, we were given a center of \((-2, 3)\) and a radius of 4. This makes \(h = -2\), \(k = 3\), and \(r = 4\).
Substituting these values into the general equation of a circle: \((x - h)^2 + (y - k)^2 = r^2\), transforms it into \( (x - (-2))^2 + (y - 3)^2 = 4^2 \). Simplifying further results in \( (x + 2)^2 + (y - 3)^2 = 16 \).
This substitution process is crucial because it takes the abstract formula and turns it into something tangible and specific to the problem at hand. It allows for clearer visualization and understanding of the circle's exact position and size.