Problem 84 - 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. $$x^{2}+y^{2}-2 y-3=0$$

Step-by-Step Solution

Answer

Circle center at (0, 1) and radius 2.

Step 1: Complete the Square

We start with the equation of the circle given by \(x^2 + y^2 - 2y - 3 = 0\). To express this in the center-radius form \((x-h)^2 + (y-k)^2 = r^2\), we need to complete the square for each variable. Since the \(x^2\) term is already a complete square, we only need to complete the square for the expression \(y^2 - 2y\).

Step 2: Completing the Y Terms

Take the \(y\) terms, \(y^2 - 2y\), and complete the square. To do this, take the coefficient of \(y\), which is \(-2\), divide by 2 to get \(-1\), and then square the result to get \(1\). Therefore, we add and subtract \(1\) from the equation:\[y^2 - 2y + 1 - 1 - 3 = 0\]

Step 3: Simplify the Equation

Rewrite the equation by simplifying the completed square and constant terms:\[x^2 + (y-1)^2 - 4 = 0\]Add 4 to both sides to isolate the square terms:\[x^2 + (y-1)^2 = 4\]

Step 4: Derive the Center and Radius

From the equation \(x^2 + (y-1)^2 = 4\), we can see it is now in the form \((x-h)^2 + (y-k)^2 = r^2\). Therefore, the center \((h, k)\) is \((0, 1)\), and the radius \(r\) is \(\sqrt{4} = 2\).

Step 5: Graphing the Circle

To graph the circle, plot the center at \((0, 1)\) on a coordinate plane. From this point, measure a radius of 2 units in all directions (up, down, left, and right) and draw the circle.

Key Concepts

Center-Radius Form

Understanding the center-radius form of a circle's equation is crucial when identifying key circle characteristics. The formula \((x-h)^2 + (y-k)^2 = r^2\) represents the center-radius form for a circle. This structure reveals two vital details:

• The circle's center, denoted as \((h, k)\).
• The circle's radius, given by \(r\).

To interpret a given circle equation, rearrange it into this form. Once done, you can easily determine both the center's coordinates and the radius. The conversion often involves algebraic manipulation, such as completing the square. By mastering this conversion, you'll unlock the circle's geometric properties, making the task of graphing much simpler.

Completing the Square

Completing the square is an algebraic technique used to transform quadratic equations into a convenient form. This process involves turning part of the equation into a perfect square trinomial. In the context of circle equations, it helps to express the equation in center-radius form.

To "complete the square" for an expression like \(y^2 - 2y\), follow these steps:

• Identify the coefficient of the linear term (here, it's \(-2\) for \(y\)).
• Halve it, resulting in \(-1\), then square it to get \(1\).
• Add and subtract this square inside the equation, keeping it balanced: \(y^2 - 2y + 1 - 1\).
• Rewrite as a perfect square: \((y-1)^2 - 1\).

Once each relevant part of the equation is in this form, combining them leads to the desired structure, facilitating the identification of the circle's center and radius.

Graphing Circles

Graphing a circle effectively ties together the algebraic and visual understanding of the problem. With the equation in the form \((x-h)^2 + (y-k)^2 = r^2\), follow these steps to draw the circle:

• First, locate the center on a coordinate plane using the values \((h, k)\).
• Then, use the radius \(r\) to measure and draw from the center.

Start at the center, moving the radius length in all four cardinal directions: up, down, left, and right. These points help outline the boundary of the circle. Connecting these points with a smooth curve completes the circle diagram. This visual representation shows the result of correctly applying the center-radius form and completing the square, emphasizing the circle's symmetry and spatial location.