Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the circle satisfying the given conditions. Center \((2,-1)\), goes through \((5,3)\)

Step-by-Step Solution

Answer

The circle's equation is \((x - 2)^2 + (y + 1)^2 = 25\).

Step 1: Recall the Standard Form of a Circle's Equation

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Identify the Center of the Circle

We are given that the center of the circle is \((2, -1)\). So, \(h = 2\) and \(k = -1\).

Step 3: Use the Distance Formula to Find the Radius

We find the radius by calculating the distance between the center \((2, -1)\) and the point \((5, 3)\) which lies on the circle. The distance formula is: \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).Substitute the values: \[r = \sqrt{(5 - 2)^2 + (3 - (-1))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]

Step 4: Substitute the Center and Radius into the Circle's Equation

Now that we have \(h = 2\), \(k = -1\), and \(r = 5\), substitute these into the standard form \((x - h)^2 + (y - k)^2 = r^2\):\[(x - 2)^2 + (y + 1)^2 = 25\]

Key Concepts

Distance Formula

The distance formula is an essential tool in geometry for finding the distance between two points in a coordinate plane. This formula is particularly useful when working with circles to determine the radius. The distance formula is mathematically expressed as:\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• \(r\) represents the distance or radius in the case of a circle.

To find the radius of the circle, this formula calculates the straight-line distance between the center of the circle and a point on its circumference. For instance, when given the center \((2, -1)\) and a point on the circle \((5, 3)\), you'd substitute into the formula, resulting in:\[ r = \sqrt{(5 - 2)^2 + (3 - (-1))^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]This tells us that the radius is 5 units. Understanding the distance formula is pivotal in calculating distances in various geometric problems.

Center of a Circle

The center of a circle is the fixed point from which all points on the circle are equidistant. Knowing the center is critical when you need to formulate the equation of a circle in its standard format.In the context of a circle's equation, the center is denoted by the coordinates \((h, k)\). This pair of numbers plays a vital role as they help define the location of the circle on a graph.

• In the problem statement, the center of the circle is \((2, -1)\).
• This means that every point on the circle is exactly the radius distance of 5 units from \((2, -1)\).

Grasping the concept of the circle's center helps to understand where the circle is located and symmetrical properties it may exhibit. The center is always the midpoint of the circle.

Equation of a Circle

The equation of a circle is one of the fundamental concepts in geometry that ties together the center and radius. This equation allows us to represent the circle mathematically in a coordinate plane.The standard form of a circle's equation is expressed as:\[ (x - h)^2 + (y - k)^2 = r^2\]Where

• \((h, k)\) is the center of the circle.
• \(r\) is the radius.

For a circle with center \((2, -1)\) and radius 5, substituting these values into the equation provides the specific equation:\[ (x - 2)^2 + (y + 1)^2 = 25\]This form becomes very useful to analyze or plot the circle on a graph. The equation of the circle is not just for plotting, but also assists in determining whether a given point lies inside, outside, or on the circle.

Geometry

Geometry is a branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. Understanding the geometry of circles specifically includes knowing how they are defined and how their properties interact. Circles are a fundamental element in geometry, merging numbers with graphical representation. When exploring the geometry of a circle, we typically deal with:

• Radius - the distance from the circle's center to any point on the perimeter.
• Diameter - twice the length of the radius.
• Circumference - the total distance around the circle.

In circles, geometry also involves applying the Pythagorean theorem while working with the circle equation. It helps balance problem-solving through visual and numerical means. The study and application of circle properties reveal deep insights into symmetry and uniformity in geometry.