Problem 11 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 14 , find the center and radius of each circle, and draw a sketch of the graph. $$ x^{2}+y^{2}-6 x-8 y+9=0 $$

Step-by-Step Solution

Answer

Center: (3, 4), Radius: 4

Step 1 - Write the Equation in Standard Form

The given equation is: \(x^{2} + y^{2} - 6x - 8y + 9 = 0\) To find the center and radius, we must write this in the standard form of a circle equation, which is: \((x - h)^{2} + (y - k)^{2} = r^{2}\)

Step 2 - Group and Complete the Square

First, group the \(x\) and \(y\) terms: \(x^{2} - 6x + y^{2} - 8y = -9\) Complete the square for both groups. For the \(x\) terms: \(x^{2} - 6x\), take half of \(-6\) to get \(-3\), then square it to get \(9\). For the \(y\) terms: \(y^{2} - 8y\), take half of \(-8\) to get \(-4\), then square it to get \(16\). Add these squares to both sides of the equation.

Step 3 - Add Completed Squares

\(x^{2} - 6x + 9 + y^{2} - 8y + 16 = -9 + 9 + 16\) Simplified, this becomes: \((x - 3)^{2} + (y - 4)^{2} = 16\)

Step 4 - Identify the Center and Radius

Now, the equation \((x - 3)^{2} + (y - 4)^{2} = 16\) is in standard form: \((x - h)^{2} + (y - k)^{2} = r^{2}\), where \((h, k)\) is the center and \(r\) is the radius. So, the center is \((3, 4)\) and the radius is \(\sqrt{16} = 4\).

Step 5 - Sketch the Graph

Plot the center at point \((3, 4)\) on the coordinate plane. Draw a circle with radius 4 units around this center. Ensure the circle is accurate by measuring 4 units in all directions from the center.

Key Concepts

standard form of a circle

To fully grasp the equation of a circle, it's important to understand its standard form. The standard form of a circle is:ewline \((x - h)^{2} + (y - k)^{2} = r^{2}\) where

• ewline The term \((x - h)\) shifts the circle horizontally by 'h' units.
ewline
• The term \((y - k)\) shifts the circle vertically by 'k' units.
ewline
• ewline The radius 'r' is the distance from the center \((h, k)\) to any point on the circle.

ewlineSo when working with the equation of a circle, your goal is to arrange it into this form because it provides direct information about the circle's center and radius. In our specific problem, the equationewlineewline \(x^{2} + y^{2} - 6x - 8y + 9 = 0\)ewlineis not initially in this standard form, which leads us to the next step: completing the square. New Line

completing the square

Completing the square is a method used to convert quadratic equations into a form that reveals insights into the geometric properties of conic sections like circles.It involves creating perfect squares from the given quadratic expressions. Let's look at the steps:

• Group the x and y terms: \((x^2 - 6x) + (y^2 - 8y)\).
• Complete the square for x-terms. Take half of \(-6\) (the coefficient of x), which is -3, and then square it to get 9. Add and subtract 9 within the equation to balance it.
• Repeat the process for y-terms. Half of \(-8\), which is -4, squared is 16. Add and subtract 16.
• Add these adjusted terms to both sides of the equation: ewlineewline \(x^2 - 6x + 9 + y^2 - 8y + 16 = -9 + 9 + 16\). This simplifies to the perfect squares: \( (x - 3)^2 + (y - 4)^2 = 16\).

At this stage, we have the standard form of the circle. This helps to derive the center (h, k) and the radius (r).

graphing circles

Once the equation is in standard form, graphing a circle becomes much easier and straightforward. Let's break it down:

• Identify the center: from the equation \((x - 3)^{2} + (y - 4)^{2} = 16\), the center is \((3, 4)\).By comparing it to \((x - h)^{2} + (y - k)^{2} = r^{2}\), we see that h=3 and k=4.
• Determine the radius: The radius r is \sqrt{16} = 4.\

To graph the circle:

• Plot the center at point \( (3, 4) \) on the coordinate plane.
• Use a compass or measure from the center to plot points 4 units away in all directions.
• Connect these points in a smooth, round shape to form the circle. Ensure symmetry to get an accurate circle.

Graphing circles helps visualize their properties and relationships to other geometric figures.Understanding these steps enhances your ability to tackle various mathematical problems involving circles.