Problem 96 - College Algebra and Calculus: An Applied Approach

Question

Write the equation of the circle in standard form. Then sketch the circle. \(5 x^{2}+5 y^{2}+10 x+1=0\)

Step-by-Step Solution

Answer

The standard form of the circle is \((x+1)^{2} + y^{2} = \frac{4}{5}\), with the center at (-1,0) and radius of \(\sqrt{\frac{4}{5}}\).

Step 1: Form grouping for x and y

Rewrite the given equation \(5x^{2}+5y^{2}+10x+1=0\) to form grouping of x's and y's. This can be done by dividing the equation by 5 and rearranging: \(x^{2}+y^{2}+2x+ \frac{1}{5}=0\) becomes \((x^{2}+2x+1) + y^{2} = -\frac{1}{5}+1\).

Step 2: Complete the square

Here, complete the square for x. Completing the square allows us to present the equation in square form (a+b)^{2}. We have \((x+1)^{2} + y^{2} = 1-\frac{1}{5}\). This equation needs to be rewritten as \((x+1)^{2} + y^{2} = \frac{4}{5}\)

Step 3: Write in standard form

Write the obtained expression in the standard form of the circle equation, \( (x-h)^{2}+(y-k)^{2}=r^{2}\). After observing the expression \((x+1)^{2} + y^{2} = \frac{4}{5}\), we have h equal to -1, k equal to 0, and +r equals to \(\sqrt{\frac{4}{5}}\).

Step 4: Sketch the circle

Plot the center of the circle at coordinates (-1,0) on a graph. Then, plot another point at a distance of the radius along any direction from the center. Draw a circle with radius \(\sqrt{\frac{4}{5}}\) around the center.

Key Concepts

Completing the Square

Completing the square is an essential algebraic technique used to transform a quadratic expression into a perfect square trinomial, making it easier to solve or graph equations, particularly circles. To complete the square, we focus on expressions of the form ax^2 + bx + c. The goal is to create a new expression, (x + p)^2, which expands to x^2 + 2px + p^2.

The process involves dividing b, the coefficient of x, by 2 to get p, and then adding and subtracting p^2 within the quadratic expression to maintain equality. We then regroup to get our perfect square trinomial and a constant. When applied to the circle's equation, it allows us to express the equation in standard form, revealing the circle's center and radius clearly.

Graphing a Circle

Graphing a circle requires understanding its geometric properties. A circle is defined as the set of all points equidistant from a fixed point, the center. To graph a circle, you need to identify its center (h, k) and radius r, which are found in the standard form of the circle's equation, (x - h)^2 + (y - k)^2 = r^2.

Start by plotting the center on a coordinate plane. Then, using the radius, mark points around the center at the distance of the radius in all directions. Connect these points with a smooth, round curve to complete the circle. The circle's equation in standard form makes plotting straightforward, as it directly provides the necessary components for graphing.

Algebraic Manipulation

Algebraic manipulation involves the skillful rearranging and simplifying of algebraic expressions and equations. This process includes numerous techniques, such as distributing, factoring, combining like terms, and using inverse operations.

For the equation of a circle, algebraic manipulation is critical in transforming the equation into a workable form. We typically start by getting all terms on one side of the equation and then isolate the terms with variables. In the step-by-step solution above, dividing the equation by 5 and rearranging the terms are forms of algebraic manipulation, setting the stage for completing the square and ultimately rewriting the equation into standard form.

Standard Form of a Circle

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 of the circle. It's an organized and widely used format because it clearly conveys the fundamental characteristics of the circle, making the analysis and graphing of the circle much more accessible.

To rewrite the equation of a circle into standard form, we often use the completing the square method to deal with the x and y terms separately, as seen in the provided example. When the equation is in standard form, it becomes straightforward to identify the precise location of the circle on the coordinate plane and its size.