Problem 100 - College Algebra and Calculus: An Applied Approach

Question

Write the equation of the circle in standard form. Then sketch the circle. . \(x^{2}+y^{2}-4 x+2 y+3=0\)

Step-by-Step Solution

Answer

The equation in standard form is \((x-2)^{2}+(y+1)^{2} = 2\). The centre of the circle is at point (2,-1) and the radius is \(\sqrt{2}\).

Step 1: Rewriting the equation

Rearrange the given equation by grouping the x terms together and the y terms together: \(x^{2}-4 x+y^{2}+2 y = -3\)

Step 2: Completing the Square

In order to complete the square, take half of the coefficient of x (which is -4), square it and add it to both sides. Do the same for y. This results in: \((x^{2}-4x+4)+(y^{2}+2y+1) = -3 + 4 + 1\)

Step 3: Formulate Standard Form

The equation \((x^{2}-4x+4)+(y^{2}+2y+1) = 2\) now becomes \((x-2)^{2}+(y+1)^{2} = 2\), which is the standard form of a circle.

Step 4: Graph the Circle

To sketch the circle, plot the center point (2,-1) on the coordinate plane. Use the radius' value, which is the square root of 2, to draw the circle. The radius can be measured up, down, left, and right from the center to draw a perfect circle

Key Concepts

Completing the Square

Completing the square is a method used in algebra to transform a quadratic equation into a perfect square trinomial. This approach is especially useful in geometry when working with the equations of circles. In the context of our exercise, we apply this technique to rewrite the circle equation in a simpler and more recognizable form.
To complete the square for the equation \(x^2 - 4x + y^2 + 2y = -3\), we follow these steps:

• For the \(x\) terms: Identify the coefficient of \(x\), which is \(-4\). Take half of \(-4\) to get \(-2\), then square it to obtain \(4\). Add and subtract this square within the equation.
• For the \(y\) terms: Identify the coefficient of \(y\), which is \(2\). Half of \(2\) is \(1\), and squaring it results in \(1\). Add and subtract this value in the equation as well.

By adding these cubic terms to both sides of the equation, we maintain equation balance and convert the expression into a more manageable form. Thus, the revised equation becomes \((x - 2)^2 + (y + 1)^2 = 2\). This method simplifies the equation, showcasing a standard form of a circle.

Standard Form of a Circle

The standard form of a circle's equation is a widely used formula in geometry that looks like \((x - h)^2 + (y - k)^2 = r^2\). Here, \((h, k)\) represents the center of the circle on a coordinate plane, and \(r\) is the radius.
In our exercise, after completing the square, we obtained \((x - 2)^2 + (y + 1)^2 = 2\). By comparing this with the standard form, it's evident:

• The center of the circle is \((2, -1)\).
• The radius squared \(r^2\) is \(2\). Hence, the radius \(r\) is the square root of \(2\).

This format makes it easy to visualize and sketch the circle on a graph. Identifying these parameters allows us to precisely draw and understand the circle's placement and size on the coordinate plane.

Coordinate Plane

A coordinate plane is a two-dimensional surface where we can plot points, lines, and shapes using a system of horizontal and vertical axes. For circles, this plane allows us to visualize the equation and locate significant points like the center and radius.
When working with the equation \((x - 2)^2 + (y + 1)^2 = 2\), the coordinate plane provides the setting to interpret this circle. Here's how it works:

• Identify the center at the point \((2, -1)\) on the plane by moving 2 units to the right and 1 unit down from the origin.
• From this center, use the radius, calculated as \(\sqrt{2}\), extending in all directions (up, down, left, right) equally.

By marking these distances, we can draw a circle accurately. The consistency of this plotting method is crucial for ensuring that the circle is proportionate and situated correctly. Understanding the principles of the coordinate plane is key to mastering graphical representations in geometry.