Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find an equation for the circle with center (0,0) and radius \(r=1\).

Short answer

The equation for the circle is \(x^2 + y^2 = 1\).

Step-by-step solution

Understand the Standard Form of Circle Equation

The standard form of the equation for a circle with center \(h, k\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\).

Identify the Center and Radius

In this problem, the center of the circle is given as (0,0) and the radius \(r\) is 1.

Substitute the Values into the Standard Form

Plug in \(h = 0\), \(k = 0\), and \(r = 1\) into the standard form equation: \((x - 0)^2 + (y - 0)^2 = 1^2\).

Simplify the Equation

After substitution, the equation simplifies to \(x^2 + y^2 = 1\).

Key concepts

Standard Form of Circle

To write an equation of a circle, we rely on its standard form. The standard form of the equation for a circle is \((x - h)^2 + (y - k)^2 = r^2\). This equation tells us everything we need to know about a circle:

• \(h\) and \(k\) represent the coordinates of the center of the circle.
• \(r\) represents the radius of the circle.

Let's break it down a bit further. Since \(h\) and \(k\) are the coordinates of the center, \(h\) shifts the circle left or right on the x-axis and \(k\) shifts it up or down on the y-axis.
The \(r\) value, squared, gives the size of the circle. When we plot this equation, we get a perfect circle centered at \( (h, k) \) with a radius of \( r \).

Radius of Circle

The radius of a circle is the distance from the center of the circle to any point on the circle. It's a key element in the circle's equation and fundamentally shapes its size. Let's break down some key points about the radius:

• It is always a positive number.
• In the equation \((x - h)^2 + (y - k)^2 = r^2\), the \(r^2\) term represents the square of the radius.
• If you are given the radius, you may need to square it to use in the standard equation.

For our specific example, where the radius \(r\) is given as 1, plugging this into the standard form equation is straightforward.
Since \(r = 1\), \(r^2\) is simply 1. This simplifies the equations considerably, giving us \((x - 0)^2 + (y - 0)^2 = 1\).
Finally, simplifying this further gives \(x^2 + y^2 = 1\), which is the simplest form of our circle equation for a radius of 1.

Center of Circle

The center of a circle is a fixed point from which every point on the circumference is equidistant. The coordinates of the center are \((h, k)\) in the standard circle equation. Here's a deeper look:

• The center is crucial in defining the position of the circle on the graph.
• When \(h\) and \(k\) are both zero, the circle is centered at the origin of the coordinate system.

In our exercise, the center is given as \((0, 0)\). This means our circle is centered exactly at the origin of our coordinate plane.
Plugging these values into our standard form equation where \(h = 0 \) and \( k = 0 \), the equation simplifies to: \((x - 0)^2 + (y - 0)^2 = 1^2\), which eventually reduces to \(x^2 + y^2 = 1\).
This shows that our circle is perfectly centered at the origin, with all points on the circle being 1 unit away from the center.