Problem 114 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The graph of the equation \(x^{2}+y^{2}=r^{2}\) will have \(x\) -intercepts \((\pm r, 0)\) and \(y\) -intercepts \((0, \pm r)\).

Step-by-Step Solution

Answer

The statement is true. The graph of the equation \(x^{2}+y^{2}=r^{2}\) will indeed have x-intercepts at \((\pm r, 0)\) and y-intercepts at \((0, \pm r)\).

Step 1: Substitute y=0 into the equation

First, replace 'y' in the equation with zero to find the x-intercepts of the circle. This will yield \(x^{2}+0^{2}=r^{2}\) simplifying to \(x^{2}=r^{2}\).

Step 2: Find the x-intercepts

This equation implies that x could be \(+r\) or \(-r\). Hence we have two x-intercepts at \((\pm r,0)\).

Step 3: Substitute x=0 into the equation

Next, replace 'x' in the equation with zero to find the y-intercepts of the circle. This will yield \(0^{2}+y^{2}=r^{2}\), simplifying to \(y^{2}=r^{2}\).

Step 4: Find the y-intercepts

This equation implies that y could be \(+r\) or \(-r\). Hence we have two y-intercepts at \((0,\pm r)\).

Key Concepts

Intercepts

Intercepts are the points where a graph crosses the axes. In the context of circle geometry, these intercepts can be found by analyzing the equation of the circle. Let's start with x-intercepts. To find where a circle crosses the x-axis, we set y to zero in the equation. For the equation \(x^2 + y^2 = r^2\), substituting y = 0 simplifies the equation to \(x^2 = r^2\). Solving for x gives us the solutions \(x = \pm r\). This means the x-intercepts are located at the points \((r, 0)\) and \((-r, 0)\).

For y-intercepts, the process is similar but inverted. We set x to zero and solve for y. Using the same circle equation \(x^2 + y^2 = r^2\) and substituting x = 0 simplifies the equation to \(y^2 = r^2\). Solving for y provides us with the solutions \(y = \pm r\). Therefore, the y-intercepts are at \((0, r)\) and \((0, -r)\).

To summarize:

• x-intercepts: \((r, 0)\) and \((-r, 0)\)
• y-intercepts: \((0, r)\) and \((0, -r)\)

These intercepts give us a clear indication of where the circle touches the axes.

Equation of a Circle

The equation \(x^2 + y^2 = r^2\) is the standard form for a circle centered at the origin of a coordinate plane with radius r. In this equation:

• \(x\) and \(y\) are variables representing coordinates of any point on the circle.
• \(r\) is a constant representing the radius of the circle, which is the distance from the center of the circle to any point on its circumference.

To understand this concept better, consider how the circle's radius influences its size. A larger radius means a larger circle, but the center remains fixed at the origin (0, 0).

This formula is derived from the Pythagorean Theorem. Any point on the circle is at a distance r from the center. The distance formula, \(\sqrt{(x - 0)^2 + (y - 0)^2}\), is simplified to \(x^2 + y^2 = r^2\) because the center is at (0, 0), making the calculation straightforward. This foundational equation allows us to explore various properties and relationships in circle geometry.

Graphing Circles

Graphing a circle given the equation \(x^2 + y^2 = r^2\) involves plotting all the points that satisfy this equation. First, identify the center and the radius. Here, we have:

• Center: \((0, 0)\)
• Radius: \(r\)

The graph of the circle will be perfectly symmetric around the origin. Every point on the circle's edge is exactly distance r from this center.

When graphing:

• Plot the center at the origin.
• Use intercept points such as \((r,0)\), \((-r,0)\), \((0, r)\), and \((0, -r)\) as references to draw the circle's outline.
• Ensure to keep equidistant from the center all around for an accurate circle representation.

Graphing a circle accurately requires understanding this relationship between the algebraic equation and its geometric representation. This skill forms the basis for more advanced concepts in coordinate geometry.