Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the circle satisfying the given conditions. Center \((-2,3)\), radius 4

Step-by-Step Solution

Answer

\((x + 2)^2 + (y - 3)^2 = 16\).

Step 1: Understand the Circle Equation Formula

The standard 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.

Step 2: Identify Parameters from the Problem

From the problem, we identify that the center of the circle \((h, k)\) is \((-2, 3)\) and the radius \(r\) is \(4\).

Step 3: Substitute Parameters into the Circle Equation

Substitute \(h = -2\), \(k = 3\), and \(r = 4\) into the circle equation formula \[ (x - (-2))^2 + (y - 3)^2 = 4^2 \] which simplifies to \[ (x + 2)^2 + (y - 3)^2 = 16 \].

Step 4: Simplify the Equation

Since the equation already appears simplified, \[ (x + 2)^2 + (y - 3)^2 = 16 \] is the final equation of the circle.

Key Concepts

Circle

A circle is a fundamental shape in geometry consisting of all points in a plane that are at a constant distance from a given fixed point, known as the center. This consistent distance is called the radius. Circles are perfect examples of symmetry and have various properties widely used in mathematics, particularly in geometry and trigonometry.

• They always have a constant radius.
• The distance from the center to any point on the circle is the same.
• Circles are defined by their center and radius, crucial for writing their equations.

Understanding circles is key to mastering many geometric and algebraic problems.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves studying geometric shapes using the coordinate plane. The coordinate plane is a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis).
In the context of circles, coordinate geometry helps in placing the circle in the plane with precision.

• The coordinates of the center \(h, k\) place the circle at the proper position on the plane.
• The radius defines how far each point on the circle is from this center.
• Coordinate geometry assists in calculating distances, slopes, and understanding the relations between various geometric entities.

This study is foundational for understanding not just circles but all geometric figures on a plane.

Radius and Center

The radius and center are fundamental properties that define a circle's location and size. In any circle, the center is the fixed point from which every point on the perimeter is equidistant, and the radius is this constant distance. Knowing these allows us to write the circle's equation and understand its properties.
For any circle:

• The center \(h, k\) provides the exact placement in the plane.
• The radius \(r\) tells us how big the circle is.
• Larger radii make larger circles.

These two terms are crucial for deriving the standard form equation of a circle.

Standard Form of Circle Equation

The equation of a circle in standard form provides a concise mathematical description of its properties. It is typically written as \((x-h)^2 + (y-k)^2 = r^2\), where \(h, k\) are the coordinates of the center and \(r\) is the radius. This equation is fundamental for solving mathematical problems involving circles.

• The left side of the equation represents the squared distances from the center along the x and y axes.
• The right side \(r^2\) represents the square of the radius, indicating how far the circle's edge is from the center.
• This form allows easy substitution of the circle's parameters to find its equation e.g., substituting \((h, k) = (-2, 3)\) and \(r = 4\) results in \((x + 2)^2 + (y - 3)^2 = 16\).

Understanding this form enables solving and composing equations for circles efficiently.