Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

A moiré pattern is formed when two geometrically regular patterns are superimposed. Shown in the figure is a pattern obtained from the family of circles \(x^{2}+y^{2}=n^{2}\) and the family of horizontal lines \(y=m\) for integers \(m\) and \(n\). (a) Show that the points of intersection of the circle \(x^{2}+y^{2}=n^{2}\) and the line \(y=n-1\) lie on a parabola. (b) Work part (a) using the line \(y=n-2\). Exercise 49

Step-by-Step Solution

Answer

The points of intersection lie on parabolas described by \(x^{2} = 2y + 1\) and \(x^{2} = 4(y + 1)\).

Step 1: Set Up the Equations

We start with the circle equation \(x^{2} + y^{2} = n^{2}\) and solve for the line \(y = n-1\). Substitute \(y = n-1\) into the circle equation. This substitution gives us \(x^{2} + (n-1)^{2} = n^{2}\).

Step 2: Simplify the Equation

Simplify the equation \(x^{2} + (n-1)^{2} = n^{2}\). Calculate \((n-1)^{2}\) as \(n^{2} - 2n + 1\). Substitute back into the equation: \(x^{2} + n^{2} - 2n + 1 = n^{2}\).

Step 3: Solve for x

Subtract \(n^{2}\) from both sides, yielding \(x^{2} = 2n - 1\). This equation describes the x-values of the intersection points.

Step 4: Recognize the Parabola

Observe that the equation \(x^{2} = 2n - 1\) can be rewritten as \(x^{2} = 2y + 1\), which is a standard form of a parabola opening horizontally.

Step 5: Repeat for Different Line

Now use \(y = n-2\) and substitute into the circle equation: \(x^{2} + (n-2)^{2} = n^{2}\). Simplify to get \(x^{2} + n^{2} - 4n + 4 = n^{2}\).

Step 6: Simplify and Solve for x (Different Line)

Subtract \(n^{2}\) from both sides, yielding \(x^{2} = 4n - 4\). This is another equation describing the x-values of intersection points.

Step 7: Identify the Parabola (Different Line)

Rewrite the result \(x^{2} = 4n - 4\) as \(x^{2} = 4(y + 1)\), which is again a parabola equation with a horizontal opening.

Key Concepts

Understanding Moiré Patterns

Moiré patterns are fascinating optical phenomena that happen when two or more sets of lines or curves overlap and create a new visual pattern. Often, these patterns appear wavy or rippled, providing a mesmerizing effect that captures our attention. This occurs because when grids of lines or curves align in a specific way, interference patterns emerge, which can be observed clearly in moiré designs.

• These patterns are usually not present in the original single structures.
• Their appearance and complexity depend on the density, angle, and shape of the overlapping structures.

By thinking about circles, lines, or even grids, we can see how moiré effects can bring various geometric concepts together. It's crucial for students to understand these interactions to visualize better how simpler components combine to produce more intricate results.

The Geometry of Circles

Circles are fundamental shapes in geometry, characterized by all points being equidistant from a center point. In the equation of a circle, such as \(x^{2} + y^{2} = n^{2}\), the circle is centered at the origin \(0,0\) with a radius \(n\). Understanding this equation helps in comprehending more complex geometric interactions.

• The equation reflects the Pythagorean theorem in two dimensions.
• The radius is a crucial element, as it determines the size of the circle.

When circles intersect with other lines or curves, like straight lines or parabolas, they reveal fascinating points of intersection, leading us to explore complex geometric solutions.

Exploring Parabolas in Geometry

Parabolas are curves described by quadratic equations. The standard form is \(y = ax^2 + bx + c\) when the parabola opens vertically. In our exercise, the parabola arises in the process of finding intersection points where the circle and line meet. For example, \(x^{2} = 2y + 1\) is an equation for a parabola opening horizontally. Recognizing this form allows us to predict the behavior and shape of the parabola.

• Parabolas have distinct features like vertex, axis of symmetry, and focus.
• They appear frequently in physics and engineering due to their natural reflective properties.

Our exercise involved translating a problem of finding where lines and circles meet into one of understanding parabolas, demonstrating the powerful link between algebra and geometry.

Finding Intersection Points

Intersection points are critical in geometry, revealing where and how different shapes meet and interact. Vertical or horizontal lines might intersect curves or circles at specific points. In analytic geometry, calculating these points often involves solving equations simultaneously.

• They help us understand relationships between different geometric figures.
• Intersection points can tell us about symmetry, tangents, and even the area of shapes.

Specifically in our task, we looked at lines \(y = n-1\) and \(y = n-2\) intersecting with circles, leading to equations that unveil the x-coordinates of intersection points. Recognizing and analyzing these points provide deeper insights into the underlying geometry and algebra.