Problem 27 - 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. $$ (x-1)^{2}+(y-3)^{2}=4 $$

Step-by-Step Solution

Answer

The circle is centered at (1, 3) with a radius of 2.

Step 1: Identify the Standard Form

The given equation \((x-1)^2 + (y-3)^2 = 4\) is in the standard form of a circle equation \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Determine Center and Radius

From the equation \((x-1)^2 + (y-3)^2 = 4\), identify the center \((h, k) = (1, 3)\) and the radius \(r = \sqrt{4} = 2\).

Step 3: Translate the Axes

To translate the graph, shift the axes. Treat \(h=1\) as the new \(X\)-axis and \(k=3\) as the new \(Y\)-axis center point. This means the origin of the new axes \((X,Y)\) is at \((1,3)\).

Step 4: Plot the Circle

Using the new origin \((X,Y)\), plot the circle with radius 2 centered at the origin \((0,0)\) or \((1,3)\) in the original coordinate system, keeping both the original and translated axes visible.

Step 5: Verify the Translation

Ensure the translated graph matches the circle equation \((X)^2 + (Y)^2 = r^2\) with the center \((0,0)\) and the shown axes allow identification of both old and new origins and scales.

Key Concepts

Equation of a Circle

A circle in mathematics is defined by a simple equation that helps identify its fundamental properties. The standard form of the equation of a circle is \[(x - h)^2 + (y - k)^2 = r^2\]where:

• \((h, k)\) represents the center of the circle.
• \(r\) is the radius, which is the distance from the center to any point on the circle.

The equation is structured as two squared terms set equal to the square of the radius. This form is very useful because it allows us to immediately see where the circle is centered and how large it is in terms of its radius. For example, in the given equation \((x-1)^2 + (y-3)^2 = 4\), it's clear that the circle's center is at \((1,3)\) and the radius \(r\) is \(2\) since \(r^2 = 4\). Understanding how to interpret and manipulate this equation is crucial for effectively dealing with problems involving circles in coordinate geometry.

Translation of Axes

The concept of translating axes involves shifting the coordinate system to make problem-solving easier. In this instance, translating the axes means adjusting the \(x\) and \(y\) coordinates such that a significant point becomes the new origin. For our circle equation, we translate the axes so that the center of the circle \((1, 3)\) becomes the origin of our new system. Therefore, we define new axes as \(X\) and \(Y\):

• For axes translation, \(X = x - 1\) and \(Y = y - 3\).

This change helps simplify the circle's equation to a basic form like \(X^2 + Y^2 = r^2\). Here, \((0,0)\) is the center in the translated coordinates, making it easier to graph without adjusting the circle's position in the original coordinate system. By using this technique, we effectively transform complicated graphing tasks into simple forms.

Coordinate Geometry

Coordinate geometry, often known as analytic geometry, combines algebra and geometry to give us a profound way to describe geometrical shapes numerically. It utilizes a coordinate plane on which each point has a unique pair of numerical coordinates \((x, y)\).

In the exercise, coordinate geometry is useful for manifesting the equation of a circle in a visual manner. By plotting the center \((1,3)\) and marking the radius of the circle as 2, we can accurately represent the circle's position and its size within the plane.

• Each point on the circle satisfies the circle's equation.
• The Cartesian plane is essential for interpreting these algebraic relationships as spatial ones.

Therefore, coordinate geometry allows us to easily visualize and solve problems related to shapes and curves, like circles, by linking algebraic expressions to geometric interpretations.

Graph Sketching

Graph sketching is a visual exercise aimed at representing equations geometrically on a coordinate plane. For a circle equation like \((x-1)^2 + (y-3)^2 = 4\), sketching involves plotting the circle while highlighting its important features:

• Determine the center from the equation — in this case, \((1,3)\).
• Calculate the radius, identified as \(2\), because the square root of \(4\) is \(2\).
• Draw the circle around its center, ensuring all points are equidistant (radius) from the center.

After translating the axes, sketch the circle with new coordinates, showing both the original \((x, y)\) and translated \((X, Y)\) axes. With practice, sketching can greatly enhance understanding by allowing one to see the geometrical interpretation of algebraic expressions, ensure calculations are correct, and visually verify graph transformations like translations.