Problem 68 - Algebra 2

Question

Graph each equation. $$ x^{2}+y^{2}=64 $$

Step-by-Step Solution

Answer

The graph of the equation \(x^2 + y^2 =64 \) is a circle with the center at the origin and radius 8.

Step 1: Identify the Type of Equation

This is an equation of a circle in the standard form \((x-h)^2 + (y-k)^2 = r^2\), with \(h\) and \(k\) being the x-coordinate and y-coordinate of the center of the circle, and \(r\) the radius of the circle. Here \(h=k=0\) as there are no constants with x and y in the equation, so the center of the circle is at the origin (0, 0).

Step 2: Calculate the Radius

In this equation, the right side gives \(r^2\) which equals 64. Hence, the radius (r) can be found by taking the positive square root of 64. This renders a radius of 8.

Step 3: Draw the Circle

With the center of the circle at the origin and the radius as 8, plot the points which are at a distance 8 from the origin in all directions. This forms the circle with equation \(x^2 + y^2 = 64\).

Key Concepts

Equation of a Circle

The equation of a circle in its standard form is:

• \((x-h)^2 + (y-k)^2 = r^2\)

Here, \( (h, k) \) represents the center of the circle, and \( r \) is the radius. This equation expresses that every point \((x, y)\) on the circle is exactly \( r \) units away from the center. In simpler terms, if you know the coordinates of the center and the radius, you can describe how to draw or identify the circle on a graph.

The given equation in the exercise is \( x^2 + y^2 = 64 \). At first glance, you'll notice that there are no \( h \) or \( k \) values, which means the center of this circle is at the origin \((0, 0)\). The circle described by this equation is perfectly symmetrical about both the x-axis and y-axis.

Radius Calculation

Calculating the radius of a circle based on its equation is straightforward. The equation typically has a form like \((x-h)^2 + (y-k)^2 = r^2\), where \( r^2 \) is a number on the right side. The most important thing is to remember that you need \( r \), not \( r^2 \).

To find \( r \), simply take the square root of the value on the right side of the equation. In the equation \( x^2 + y^2 = 64 \), the right side is 64. Hence, the radius \( r \) is \( \sqrt{64} = 8 \).

A good practice is to always ensure that you're using the positive square root when dealing with distances like radii, as these are measured in positive numbers.

Center of a Circle

The center of a circle is a critical component needed to draw the circle accurately. In the formula \((x-h)^2 + (y-k)^2 = r^2\), the center is represented by the coordinates \((h, k)\).

In many problems, particularly those focused on simpler circles, these values are zero, such as with \( x^2 + y^2 = 64 \), indicating that the circle's center is at the origin \((0, 0)\). This signifies that both the x and y coordinates of the center are zero, meaning the circle is centered where the axes intersect.

If you come across equations where \( h \) and \( k \) aren't zero, the circle will be shifted horizontally and/or vertically. Always pay close attention to these values as they determine the precise location of your circle on the coordinate plane.