Problem 43 - Intermediate Algebra

Question

Write an equation of the circle with the given center and radius. $$(2,3) ; 6$$

Step-by-Step Solution

Answer

The equation is \((x-2)^2 + (y-3)^2 = 36\).

Step 1: Understand the Standard Equation of a Circle

The standard equation of a circle in the coordinate plane is: \[(x-h)^2 + (y-k)^2 = r^2\] where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Identify Circle Parameters

From the given information, the center of the circle \((h,k)\) is \((2,3)\) and the radius \(r\) is 6. These are the values that will be substituted into the standard equation of a circle.

Step 3: Substitute Parameters into the Formula

Substitute \(h = 2\), \(k = 3\), and \(r = 6\) into the circle equation: \[(x-2)^2 + (y-3)^2 = 6^2\]

Step 4: Simplify the Equation

Simplify the right-hand side of the equation by squaring the radius: \[(x-2)^2 + (y-3)^2 = 36\] This is the equation of the circle with center \((2,3)\) and radius 6.

Key Concepts

Standard Equation of a Circle

When working with circles in coordinate geometry, the standard equation of a circle is your go-to formula. This equation is written as \[ (x-h)^2 + (y-k)^2 = r^2 \] where

• \((h, k)\) represents the center of the circle.
• \(r\) stands for the radius of the circle.

The equation showcases how each point \((x, y)\) on the circle is equidistant from the center \((h, k)\), with that distance being exactly the radius \(r\).
A key point here is that both \(h\) and \(k\) are subtracted from \(x\) and \(y\) respectively, which indicates the shift of the circle from the origin to its center. Understanding this setup makes identifying circle parameters much simpler.

Circle Parameters

Circle parameters refer to the key characteristics that define the circle on a coordinate plane: the center and the radius.
To find these, look at the parts \((h, k)\) and \(r\) from the circle's equation:

• The **center** of the circle is \((h, k)\). In this particular case, it is \((2, 3)\).
• The **radius** is \(r\). This is given as 6 in the exercise.

These parameters are crucial because they help us personalize the standard equation to represent any circle. Knowing the center and radius, we substitute them directly into the standard format, giving us an equation that precisely defines the circle's location and size on the coordinate plane.

Coordinate Geometry

In coordinate geometry, we use algebraic equations to study geometric figures. A circle's equation, as a classic example, helps us pinpoint and describe its location in a coordinate system.
When given a specific center \((h, k)\) and radius \(r\), we can easily build the circle's equation.
This represents a perfect balance, relating algebraic expressions to geometric shapes.

• The equation \((x-h)^2 + (y-k)^2 = r^2\) directly correlates to how we visualize the circle on a graph.
• This method ensures precise calculation and representation, leaving little room for overlap or misunderstanding.

Using this approach, coordinate geometry allows us not only to see but also to mathematically prove and describe the attributes of circles and other shapes within this structured plane.