Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 11-16, find the equation of the circle satisfying the given conditions. Center \((1,1)\), radius 1

Step-by-Step Solution

Answer

The equation is \((x-1)^2 + (y-1)^2 = 1\).

Step 1: Identify the Standard Equation of a Circle

The equation of a circle with center \((h, k)\) and radius \(r\) is given by the formula: \((x-h)^2 + (y-k)^2 = r^2\).

Step 2: Substitute the Given Center into the Equation

The center of the circle is \((1,1)\), so substitute \(h = 1\) and \(k = 1\) into the equation. This gives us: \((x-1)^2 + (y-1)^2 = r^2\).

Step 3: Substitute the Given Radius into the Equation

The radius of the circle is \(1\), so substitute \(r = 1\) into the equation. Now the equation is: \((x-1)^2 + (y-1)^2 = 1^2\).

Step 4: Simplify the Equation

Calculating \(1^2\) results in \(1\). So, the final equation of the circle becomes: \((x-1)^2 + (y-1)^2 = 1\).

Key Concepts

Standard Form of a Circle

When we talk about the equation of a circle, we often refer to its standard form. The standard form of a circle's equation provides us with valuable information about the circle's position and size on a coordinate plane. It's expressed as

• \((x-h)^2 + (y-k)^2 = r^2\) where:
• \(h\) and \(k\) are the coordinates of the center of the circle.
• \(r\) is the radius of the circle.

This equation can tell us everything we need to know about a circle’s location and dimensions just at a glance.
To use it, replace \(h\) and \(k\) with the x and y coordinates of your circle's center, and \(r\) with its radius. This form naturally simplifies the process of understanding and working with circles in geometry.

Circle Center

The circle center is like the heart of the circle. It’s the point from which all points on the circle are exactly the same distance away. In our standard form equation \((x-h)^2 + (y-k)^2 = r^2\),

• \((h, k)\) represents the center of the circle.

For example, if you’re given a center at \((1, 1)\), this means that point \((1, 1)\) lies directly in the middle of the circle.
Understanding the role of the circle's center in the equation makes it possible to correctly position the circle on a graph. The center coordinates are subtracted from \(x\) and \(y\) within the equation, reflecting their central role in determining the circle's placement in the coordinate plane.

Circle Radius

The radius of a circle is a crucial measurement representing the distance from the circle's center to any point on its edge. In the standard form equation \((x-h)^2 + (y-k)^2 = r^2\),

• \(r^2\) is the radius squared.

This implies that your calculated value for \(r\) simply needs to be squared and matched to whatever the square of the distance from the edge to the center might be.
For instance, if the radius is given as 1, you plug in \(r = 1\) into the equation to show that every point on the circle is one unit away from the center. Understanding this concept helps when determining the size of the circle and its relative position not just visually, but mathematically.