Question

In Problems \(13-24,\) write the standard form of the equation and the general form of the equation of each circle of radius \(r\) and center \((h, k)\). Graph each circle. $$ r=2 ;(h, k)=(0,0) $$

Short answer

The standard form is \(x^2 + y^2 = 4\) and the general form is \(x^2 + y^2 - 4 = 0\).

Step-by-step solution

Identify the given values

The problem provides the radius of the circle, which is \(r = 2\), and the center of the circle, which is \((h, k) = (0, 0)\).

Write the standard form of the equation

The standard form of the equation of a circle with radius \(r\) and center \((h, k)\) is given by: \[(x - h)^2 + (y - k)^2 = r^2\]. Substituting the given values, \(h = 0\), \(k = 0\), and \(r = 2\), we get: \[(x - 0)^2 + (y - 0)^2 = 2^2\].Simplify this to obtain the standard form: \[x^2 + y^2 = 4\].

Convert to the general form

The general form of the equation of a circle is written as: \[Ax^2 + Ay^2 + Dx + Ey + F = 0\]. Since we already have \(x^2 + y^2 = 4\), we can rearrange it to match the general form: \[x^2 + y^2 + 0x + 0y - 4 = 0\].Thus, the general form is: \[x^2 + y^2 - 4 = 0\].

Graph the circle

Plot the center of the circle at the point \((0, 0)\). Then, from the center, measure a distance of 2 units in all directions (up, down, left, and right) to form a circle. This gives us the graph of the circle.

Key concepts

radius of a circle

The radius of a circle is the distance from the center to any point on the circle. Think of it as a line segment from the exact middle to the edge of the circle. If you know this distance (radius), it is easy to understand how large the circle is. In our example, the radius is given as r = 2. This means each point on the circle is exactly 2 units away from the center (0, 0).

standard form of circle equation

The standard form of a circle's equation is very useful and easy to remember. It's written as \((x - h)^2 + (y - k)^2 = r^2\). Here, \(h\) and \(k\) represent the coordinates of the center of the circle, and \(r\) is the radius. By substituting the given values \(h = 0\), \(k = 0\), and \(r = 2\) into the formula, we get:\[ (x - 0)^2 + (y - 0)^2 = 2^2 \]Which simplifies to:\[ x^2 + y^2 = 4 \]Understanding this form allows you to quickly identify the center and radius just by looking at the equation.

general form of circle equation

The general form of a circle's equation is different from the standard form but represents the same circle. It's written as: \Ax^2 + Ay^2 + Dx + Ey + F = 0\. To convert the standard form \ x^2 + y^2 = 4 \ to the general form, we need to rearrange it. We want it to look like :\( x^2 + y^2 + 0x + 0y - 4 = 0\). Therefore, the general form for our equation is:\[ x^2 + y^2 - 4 = 0 \]. This form is beneficial when dealing with equations in algebraic operations or comparing different circle equations.

graphing circles

Graphing a circle is easy once you know the center and the radius. Start by plotting the center point, in our case, it's (0, 0). Next, measure the radius distance in all four directions (up, down, left, and right) from the center. For our example with a radius of 2, you'd plot points 2 units away from the center. Connect these outer points in a smooth, round curve to form the circle. Your graph should resemble a perfect circle centered at (0, 0) and with a radius of 2.