Problem 28 - Calculus with Concepts in Calculus

Question

Sketch the graph of the given equation with the help of a suitable translation. Show both the \(x\) and \(y\) axes and the \(X\) and \(Y\) axes. $$ (x+2)^{2}+(y+4)^{2}=\frac{1}{4} $$

Step-by-Step Solution

Answer

The circle has center (-2, -4) and radius 0.5, translated using axes \(X = x+2\) and \(Y = y+4\).

Step 1: Recognize the Standard Form

The equation \((x+2)^2 + (y+4)^2 = \frac{1}{4}\) represents a circle. The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\). In this equation, \(h = -2\), \(k = -4\), and the radius \(r = \frac{1}{2}\) (since \(r^2 = \frac{1}{4}\)).

Step 2: Determine the Center of the Circle

The center of the circle in the given equation \((x+2)^2 + (y+4)^2 = \frac{1}{4}\) is at the point \((-2, -4)\). This is because the expression is in the form \((x - (-2))^2 + (y - (-4))^2 = r^2\).

Step 3: Identify the Translations

The translations move the origin to the circle's center. Hence, new axes are defined: \(X = x+2\) and \(Y = y+4\). Thus, the translated equation for the circle in terms of \(X\) and \(Y\) coordinates becomes \(X^2 + Y^2 = \frac{1}{4}\).

Step 4: Sketch the Circle and Translations

Draw the original \(x\)- and \(y\)-axes. The circle is centered at \((-2, -4)\) on the \(xy\)-plane with a radius of \(\frac{1}{2}\). Then draw the new \(X\)- and \(Y\)-axes through the center \((-2, -4)\). The circle looks the same with respect to the \(XY\)-axes as it would around the origin in simpler cases.

Key Concepts

Translation of Axes

When we talk about the translation of axes in geometry, we're moving the origin from one point to another. This concept is particularly handy when trying to simplify complex equations or graph them. Here, the original equation

• \((x+2)^2 + (y+4)^2 = \frac{1}{4}\)

represents a circle in the \(xy\)-coordinate system. By translating the axes, we're making it more straightforward to sketch.

To translate, we redefine our axes so that the center of the circle becomes the new origin. It involves shifting both the \(x\) and \(y\) values:

• \(X = x + 2\): This shifts the \(x\)-coordinate 2 units to the left.
• \(Y = y + 4\): This shifts the \(y\)-coordinate 4 units downwards.

Now the equation appears as \(X^2 + Y^2 = \frac{1}{4}\), making it look like a simple equation of a circle centered at the origin in this new coordinate system.

Graphically, this means we draw new perpendicular axes (\(X\) and \(Y\)) passing through the circle's center. The circle maintains its shape and form relative to these new axes.

Standard Form of a Circle

Understanding the standard form of a circle's equation is foundational for graphing circles properly. The standard form is represented as:

• \((x - h)^2 + (y - k)^2 = r^2\)

where \((h, k)\) is the circle's center, and \(r\) is its radius. This form helps in quickly identifying key characteristics of the circle without any calculations.

In our example equation

• \((x+2)^2 + (y+4)^2 = \frac{1}{4}\)

we can see it conforms to the standard form with \(h = -2\) and \(k = -4\). This means the circle is centered at \((-2, -4)\).

The right side of the equation, \(rac{1}{4}\), gives us \(r^2\). Taking the square root of this, we arrive at the radius \(r=\frac{1}{2}\). Using the standard form makes it easy to interpret how a circle will look in any given coordinate plane.

Circle's Center and Radius

To understand a circle graph, determining its center and radius is crucial. The center is the point around which all points on the circle are equidistant. In the equation

• \((x+2)^2 + (y+4)^2 = \frac{1}{4}\)

the center can be found by identifying the values subtracted from \(x\) and \(y\). Here, those values are \(h = -2\) and \(k = -4\), placing the center at \((-2, -4)\).

The radius is more straightforward. It's computed from the right side of the equation. Since \(r^2 = \frac{1}{4}\), taking the square root gives us \(r = \frac{1}{2}\).

The center and radius tell us everything about the circle's position and size. So once identified, the circle can be drawn accurately, whether in traditional \(xy\) coordinates or after a translation to \(XY\) coordinates where the circle's center aligns with the origin.