Problem 83 - Algebra and Trigonometry

Question

Exercises \(81-83\) will help you prepare for the material covered in the first section of the Complete the square and write the circle's equation in standard form: $$x^{2}+y^{2}-2 x+4 y=4$$ Then give the center and radius of the circle and graph the equation.

Step-by-Step Solution

Answer

The standard form of given circle's equation is \((x - 1)^2 + (y + 2)^2 = 3^2\), the center of the circle is (1, -2) and the radius of the circle is 3.

Step 1: Rewrite the Circle’s Equation with Grouped Terms

Rearrange the equation to group the x terms and y terms together: \(x^2-2x + y^2 + 4y = 4\)

Step 2: Complete the Squares

To convert the equation into standard form, it's necessary to complete the squares for both x and y. For x, take half the coefficient of x, square it, and then add and subtract it inside the equation. Same for y:\(x^2 - 2x + 1 = y^2 + 4y + 4 - 4 + 1\)This simplifies to:\((x - 1)^2 + (y + 2)^2 = 9\)

Step 3: Determine Center and Radius

From the standard form equation of a circle \((x - h)^2 + (y - k)^2 = r^2\), the center of the circle is given by the coordinates \(h, k\) and the radius is \(r\). For the equation \((x - 1)^2 + (y + 2)^2 = 9\), the center of the circle is at \(1, -2\) and the radius is \(\sqrt{9} = 3.\)

Step 4: Graph the Equation

To graph the equation, draw a circle on a coordinate plane with the center at (1,-2) and a radius of 3

Key Concepts

Complete the Square

The method of 'completing the square' is essential in converting the equation of a circle into its standard form. This technique involves manipulating the equation so that each variable forms a perfect square binomial. To complete the square, you need to focus on the coefficients of the linear terms.

For a general term like \(ax^2 + bx\), you divide \(b\) by 2, square the result, and add it to both sides of the equation. For a term like \( -2x \), this means adding and subtracting \(1\) because \(\frac{-2}{2}\) is \( -1 \) and \( (-1)^2 = 1\). This technique creates a perfect square trinomial, which can be rewritten as a binomial squared. After repeating this process for both \(x\) and \(y\) terms in our circle equation, we transform it into an equation with clear indications of the circle's center and radius.

Circle's Center and Radius

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

After completing the square, the equation becomes a clear representation where the values of \(h\) and \(k\) directly give us the coordinates of the center. The radius of the circle is the square root of the constant term on the right-hand side of the equation. For the equation \( (x - 1)^2 + (y + 2)^2 = 9 \), it is evident that the circle has its center at \( (1, -2) \) and a radius of \(3\), since \(\sqrt{9} = 3\).

Graphing a Circle

Once we have the circle's equation in standard form and identified the center and radius, graphing the circle on a coordinate plane is straightforward.

Start by plotting the center of the circle at the coordinates found previously. From this point, use the radius to mark points that are equally distant from the center in all directions. Connect these points with a smooth, round curve to form the circle. In our exercise, we would plot the center at \( (1, -2) \) and draw a circle with a radius that extends \(3\) units in all directions, completing our graph.

Coordinate Plane

The coordinate plane is an essential tool in graphing geometric figures like circles. It consists of two perpendicular number lines that intersect at a central point known as the origin.

These number lines are the \(x\)-axis, which runs horizontally, and the \(y\)-axis, which runs vertically. Each point on this plane is represented by a pair of coordinates \( (x, y) \), indicating its distance from the origin along each axis. Graphing on a coordinate plane allows us to visualize equations and their geometrical implications, such as the size and position of a circle.