Problem 77 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form of the circle with the given equation. Determine the coordinates of the center, find the radius, and graph the circle. $$x^{2}+y^{2}+2 x+2 y-23=0$$

Step-by-Step Solution

Answer

Center: (-1, -1), Radius: 5, Circle can be graphed with these values.

Step 1: Identify the Circle Equation

The given equation is \(x^2 + y^2 + 2x + 2y - 23 = 0\). This equation is in the general form of a circle equation and needs to be converted to center-radius form.

Step 2: Rearrange and Group Terms

Rearrange the terms so similar terms are grouped together: \( (x^2 + 2x) + (y^2 + 2y) = 23 \).

Step 3: Complete the Square for x

Take the expression for \(x\): \(x^2 + 2x\). Add and subtract 1 (the square of half the coefficient of \(x\)) to complete the square: \(x^2 + 2x + 1 - 1 = (x+1)^2 - 1\).

Step 4: Complete the Square for y

Similarly, take the expression for \(y\): \(y^2 + 2y\). Add and subtract 1 to complete the square: \(y^2 + 2y + 1 - 1 = (y+1)^2 - 1\).

Step 5: Substitute and Simplify

Substitute the completed squares back into the equation: \((x+1)^2 - 1 + (y+1)^2 - 1 = 23\). Simplify it to \((x+1)^2 + (y+1)^2 = 25\).

Step 6: Identify the Center and Radius

The equation \((x+1)^2 + (y+1)^2 = 25\) is now in the center-radius form, where the center is \((-1, -1)\) and the radius is \(\sqrt{25} = 5\).

Step 7: Graph the Circle

To graph the circle, plot the center at \((-1, -1)\), and use the radius of 5 to draw the circle around this center.

Key Concepts

Center-Radius Form

The center-radius form of a circle's equation is a more intuitive way to understand and work with circles in the coordinate plane. This form is expressed as \((x-h)^2 + (y-k)^2 = r^2\), where:

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

Converting an equation from the general quadratic form, \(Ax^2 + Ay^2 + Dx + Ey + F = 0\), into this form requires completing the square. By arranging the equation into the center-radius form, one can easily identify the center and the radius, making it simpler to visualize and graph the circle on a coordinate plane.

Completing the Square

"Completing the square" is a mathematical technique used to convert a quadratic expression into a perfect square trinomial. This method is key in rearranging a circle's equation to the center-radius form. To complete the square:

• Focus on one variable at a time, either \(x\) or \(y\).
• The expression \(x^2 + bx\) becomes \((x + \frac{b}{2})^2 - (\frac{b}{2})^2\).
• This process must be applied to both \(x\) and \(y\) terms.

In the context of the original exercise, you applied this method by transforming \((x^2 + 2x)\) to \((x+1)^2 - 1\) and \((y^2 + 2y)\) to \((y+1)^2 - 1\). This reshaping helps pinpoint the circle's center and radius in a visual and computationally accessible manner.

Graphing Circles

Graphing a circle with a given equation in center-radius form is a straightforward process. Here's how you can place it on the coordinate plane:

• Identify the center \((h, k)\) and the radius \(r\) from the equation \((x-h)^2 + (y-k)^2 = r^2\).
• Plot the center point \((h, k)\) on the plane.
• Using the radius \(r\), mark points \(r\) units from the center in all directions —up, down, left, and right.
• Sketch the circle through these points, ensuring it remains equidistant from the center.

In our exercise, with a center at \((-1, -1)\) and a radius of 5, you would plot the center and then use a compass or a consistent method to mark and connect points 5 units away in all directions. This visual representation shows how centered circles manifest in a Cartesian system.