Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center } C(-4,1) \text {, radius } 3 $$

Step-by-Step Solution

Answer

The equation of the circle is \( (x + 4)^2 + (y - 1)^2 = 9 \).

Step 1: Recall the Circle Equation Formula

The general equation for a circle with center \( C(h,k) \) and radius \( r \) is:\[ (x - h)^2 + (y - k)^2 = r^2 \]

Step 2: Substitute the Center Coordinates

The problem states that the center of the circle \( C \) is \( (-4,1) \). Substitute \( h = -4 \) and \( k = 1 \) into the equation:\[ (x + 4)^2 + (y - 1)^2 = r^2 \]

Step 3: Substitute the Radius

The radius \( r \) is given as \( 3 \). Substitute \( r = 3 \) in the equation:\[ (x + 4)^2 + (y - 1)^2 = 3^2 \]

Step 4: Simplify the Radius Squared

Calculate \( 3^2 \), which is \( 9 \), to complete the circle's equation:\[ (x + 4)^2 + (y - 1)^2 = 9 \]

Key Concepts

Center of a Circle

The center of a circle is like the heart of its geometric form. This point is crucial because it defines the circle's position in the coordinate plane. A circle is composed of all the points that are equidistant from this central point.

In a coordinate system, the center is represented as a point \( C(h, k) \), where \( h \) is the x-coordinate, and \( k \) is the y-coordinate.
These coordinates dictate precisely where the circle sits on the graph.

• For example, a center \( C(-4, 1) \) means that point is four units to the left of the origin and one unit up.

Once you know the center, you can readily progress to find other properties of the circle.

Radius of a Circle

The radius of a circle is a line segment that joins the center to any point on the circle itself.
The radius serves as a measurement of the circle's size, and the distance is always constant from the center to the edge.

• In mathematical terms, if the radius is given as \( r = 3 \), then every point on the circle is exactly 3 units away from the center.

The radius is the one value that stretches the circle outwards, affecting its appearance but not its fundamental shape. By knowing the radius, you can calculate other vital properties like the circumference and area, though in a geometric equation, we need it simply for determining the size of the circle.

Geometric Equations

Equations allow us to visually and numerically comprehend shapes, such as circles, on a graph.
The equation of a circle is a geometric representation that provides essential details like the center and radius directly from its formula.
For a given circle with center \( C(h, k) \) and radius \( r \), the equation is given by:

• \[ (x - h)^2 + (y - k)^2 = r^2 \].

This formula is crucial because:

• \( (x - h)^2 \) and \( (y - k)^2 \) represent squared distances on the x and y axes.
• \( r^2 \) is the radius squared, giving us the boundary of points that are \( r \) units away from the center.

Inserting the center and radius into the formula, such as \( C(-4, 1) \) with radius \( 3 \), provides a complete equation that showcases the circle's size and position on the plane: \[ (x + 4)^2 + (y - 1)^2 = 9 \] as all points satisfying this equation lie on the circle.