Problem 19 - Contemporary Precalculus

Question

Identify the conic section whose equation is given and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$4 x^{2}+4 y^{2}=1$$

Step-by-Step Solution

Answer

Answer: The conic section is a circle with center (0, 0) and radius $$\frac{1}{2}$$.

Step 1: Rewrite the equation in standard form

We are given the equation $$4 x^{2}+4 y^{2}=1$$, and we want to rewrite it in standard form for circles or ellipses. To do this, divide both sides of the equation by 4. $$\frac{4 x^{2}}{4}+\frac{4 y^{2}}{4}=\frac{1}{4}$$ Simplify: $$x^{2}+y^{2}=\frac{1}{4}$$

Step 2: Identify the conic section type

Now, let's compare our equation with standard forms for circles and ellipses to determine the type of conic section. Standard form for a circle: $$(x-h)^{2}+(y-k)^{2}=r^{2}$$ Standard form for an ellipse: $$\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$$ Looking at our equation ($$x^{2}+y^{2}=\frac{1}{4}$$), we can see that it matches the standard form for a circle, where \(h = 0\), \(k = 0\), and $$r^{2} = \frac{1}{4}$$. Therefore, this conic section is a circle.

Step 3: Calculate center and radius

Since our conic section is a circle, we can find its center and radius using the equation $$x^{2}+y^{2}=\frac{1}{4}$$. Center: \((h, k) = (0, 0)\) (since there are no constants in the equation) Radius: To find the radius, we need to calculate the square root of $$r^{2} = \frac{1}{4}$$. $$r = \sqrt{\frac{1}{4}}$$ $$r = \frac{1}{2}$$ So the radius is $$\frac{1}{2}$$. To summarize: - Conic section type: Circle - Center: (0, 0) - Radius: $$\frac{1}{2}$$ Now, you can easily draw the graph of this circle, knowing its center (0,0) and radius of $$\frac{1}{2}$$.

Key Concepts

Circle

A circle is a simple and symmetrical shape representing all points equidistant from a central point. In geometry, this distance is known as the radius. The circle's defining feature is its uniformity and lack of corners or edges. This makes it one of the most fundamental shapes in mathematics and an essential part of conic sections.

When dealing with circles, it is crucial to understand the central point or the center, around which all points of the circle revolve. Every point on the circle's circumference is equidistant from this central point, making circles pivotal in various mathematical and practical applications. They appear frequently in nature, technology, and architecture due to their inherent simplicity and beauty.

Center and Radius

In the context of circles, the **center** is the fixed point from which every point on the circle is equidistant. For the equation \[ x^2 + y^2 = \frac{1}{4} \] there are no variables shifted away from zero, indicating that the center is at the origin, (0, 0).

The **radius** is the distance from the center to any point on the circle. It represents how 'wide' the circle is and can be thought of as a measurement of the circle's size. For example, in the standard form equation \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( h \) and \( k \) are the center coordinates and \( r \) is the radius.
For our equation, \( r^2 = \frac{1}{4} \). By calculating the square root, we find \( r = \frac{1}{2} \).

Understanding the center and radius is essential for drawing the circle accurately and recognizing its properties.

Equation in Standard Form

Conic sections, like circles, have standard forms which help easily identify their properties. For a circle, this standard form is expressed as \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( (h, k) \) is the center and \( r \) is the radius.

This equation simplifies to \( x^2 + y^2 = r^2 \) when the circle's center is at the origin (0,0). In the exercise provided, after dividing the original equation by 4, it appears as \( x^2 + y^2 = \frac{1}{4} \). This reveals that the center is at the origin, and the radius squared \( r^2 \) is \( \frac{1}{4} \).

Using the standard form makes it easier to recognize the circle's characteristics and draw it accurately.

Graphing Conic Sections

Graphing a circle involves plotting all the points that satisfy its equation on a coordinate plane. Knowing the center and radius helps in sketching the circle with precision.

To graph a circle from an equation like \( x^2 + y^2 = \frac{1}{4} \), first identify the center point, which is at (0, 0) in this case. Then, the radius is determined to be \( \frac{1}{2} \).

With these pieces of information:

• Start at the center (0, 0).
• Plot points in all directions \( \frac{1}{2} \) units away from the center.
• Join these points smoothly to form the circle.

Graphing becomes an intuitive exercise when the circle's parameters are clearly understood, enabling quick and accurate visualization.