Problem 36 - Algebra 2

Question

Write the equation of the circle that passes through the given point and has a center at the origin. (Hint: You can use the distance formula to find the radius.) $$ (0,-3) $$

Step-by-Step Solution

Answer

The equation of the circle that passes through the point (0,-3) and has its center at the origin is \(x^2 + y^2 = 9\).

Step 1: Identify the center and the given point

The center of the circle is at the origin (0,0), and the given point is (0,-3).

Step 2: Compute the radius

The radius of the circle is the distance from the center to any point on the circle. In this case, it's the distance from the origin (0,0) to the point (0,-3), which is just 3 (as we can count the units along the y-axis).

Step 3: Write down the equation of the circle

The standard equation of a circle with center at origin is \(x^2 + y^2 = r^2\). After substituting the value of the radius into the standard equation, we get \(x^2 + y^2 = 3^2\).

Key Concepts

Center of a Circle

The center of a circle is a crucial point in understanding the geometry of the circle itself. It represents the position exactly in the middle of the circle from which all points on the circle are equidistant. This distance is what we call the radius. In many cases, the center of a circle can determine the entire structure of the circle.

When we say the center is at the origin, specifically at coordinates (0,0), it makes calculations easier. Why? Because both the x and y coordinates are zero. This simple setup allows straightforward calculations for determining various properties of the circle, such as the radius and the standard circle equation.

Start by being comfortable identifying the center as a point, usually given in ordered pair notation like (h,k). This will help you use the circle's equation effectively.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. It's one of the simplest yet most significant properties of the circle. The radius helps define the size of the circle and appears directly in the equation of a circle as a squared term, \(r^2\).

If you know the center of the circle and any other point on the circle, you can calculate the radius using the distance formula. In problems where the center is at the origin, the radius is just the absolute difference between the center's and the point's relevant coordinates. For instance, if the point is (0,-3), simply count the number of units along the y-axis from (0,0), which gives a radius of 3 in this example.

This concept is fundamental when writing the standard form of a circle's equation.

Distance Formula

The distance formula is a powerful tool in geometry to find the distance between two points on a coordinate plane. It's derived from the Pythagorean theorem and is written as:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

For a circle-related problem, knowing this formula helps find the radius. When dealing with a circle centered at the origin, say (0,0), the formula simplifies since one of the points is always (0,0). You only need to consider the coordinates of the other point on the circle.

In our exercise, the point (0,-3) makes calculating the distance effortless. Simply subtract the y-coordinates, obtain \((0 - (-3))\), which leads to the distance, or radius \(r = 3\).

Mastering the distance formula will allow you to solve any problem that involves finding lengths between points, including radii of circles.