Problem 35 - 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.) $$ (0,4) $$

Step-by-Step Solution

Answer

The equation of the circle that passes through (0,4) and has a center at the origin is \( x^2 + y^2 = 16 \).

Step 1: Calculate the distance from the origin to the given point

That distance will be the radius (r) of the circle. The formula for distance between two points A(x1, y1) and B(x2, y2) in a 2D plane is \( \sqrt{{(x2-x1)}^2 + {(y2-y1)}^2} \). Here, point A is the origin (0,0) and point B is (0,4). So, the radius \( r = \sqrt{{(0-0)}^2 + {(4-0)}^2} = 4 \).

Step 2: Write the equation of the circle

The general equation of a circle with center at the origin is \( x^2 + y^2 = r^2 \). Substituting the radius r as 4, the equation becomes \( x^2 + y^2 = 4^2 = 16 \).

Key Concepts

Distance Formula

To find the radius of a circle when its center is at the origin and a point on the circle is given, you use the distance formula. This formula calculates the distance between two points in a 2D coordinate system. If you have two points, \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the distance \( d \) between them can be found using:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

When the center of the circle is at the origin (0,0), and the point on the circle is (0,4), you substitute these values into the formula. Since the x-coordinates are the same, it simplifies to \( d = \sqrt{(0-0)^2 + (4-0)^2} = 4 \).
Therefore, the distance from the origin to the point (0,4) is 4, which is the radius of the circle.

Radius Calculation

Once you've used the distance formula, calculating the radius becomes straightforward. For a circle centered at the origin, the radius is simply the distance from the origin to any point on the circle. For our example, the radius is 4, as computed from the previous section.
Using the distance formula result:

• The radius \( r = 4 \)

Knowing the radius is essential because it allows you to confidently write the circle's equation. Remember, a precise radius provides clarity in mathematical equations and visual imagery in 2D geometry.

Center at Origin

A circle with its center at the origin of a coordinate plane has a unique and simplified equation. Typically, a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\) where \((h, k)\) is the center. But if the circle is centered at the origin (0,0), the formula simplifies greatly:

• The equation becomes \( x^2 + y^2 = r^2 \)

In this scenario, because the radius is 4, the equation is put into the form \( x^2 + y^2 = 16 \).
Understanding this simplification is crucial for solving problems quickly and accurately. All calculations align neatly, avoiding extra steps.

2D Geometry

Circles are an essential part of 2D geometry. They provide a means to explore spatial relationships and distances. In 2D geometry, a circle is defined by its center point and its radius. With these two components:

• The position and size of the circle are determined.
• Equations represent the circle's boundary.

An easily comprehensible circle is one centered at the origin with a simple equation such as \( x^2 + y^2 = 16 \). In two-dimensional geometry, using the origin as a center simplifies calculations and enhances problem-solving efficiency. Familiarity with the geometry of standard shapes paves the way to tackling more complex geometrics configurations.