Problem 41 - Thomas Calculus

Question

In Exercises \(41-46,\) find an equation for the circle with the given center \(C(h, k)\) and radius \(a\) . Then sketch the circle in the \(x y\) -plane. Include the circle's center in your sketch. Also, label the circle's \(x\) - and \(y\) -intercepts, if any, with their coordinate pairs. $$ C(0,2), \quad a=2 $$

Step-by-Step Solution

Answer

The circle's equation is \(x^2 + (y-2)^2 = 4\) with intercepts at \((0, 0)\) and \((0, 4)\).

Step 1: Understand the Circle Equation

The general equation of a circle with center \((h, k)\) and radius \(a\) is \((x - h)^2 + (y - k)^2 = a^2\). Here, the center is \((0, 2)\) and the radius is \(2\), so we plug these values into the formula.

Step 2: Plug Values into the Equation

Replace \(h\) with \(0\), \(k\) with \(2\), and \(a\) with \(2\) in the general circle equation: \((x - 0)^2 + (y - 2)^2 = 2^2\).

Step 3: Simplify the Equation

Simplify the equation: \(x^2 + (y - 2)^2 = 4\). This is the equation of the circle with center \((0, 2)\) and radius \(2\).

Step 4: Identify x-intercepts and y-intercepts

The \(x\)-intercepts occur when \(y = 0\): \(x^2 + (0-2)^2 = 4\). Solve for \(x\): \(x^2 + 4 = 4\), thus \(x = 0\). So, there are no \(x\)-intercepts. For \(y\)-intercepts, set \(x = 0\): \(0^2 + (y-2)^2 = 4\), so \((y-2)^2 = 4\), giving \(y = 0\) or \(y = 4\). Thus, the \(y\)-intercepts are \((0, 0)\) and \((0, 4)\).

Step 5: Sketch the Circle

Draw a coordinate plane, plot the center of the circle at \((0, 2)\), and draw the circle with radius \(2\). Mark the intercepts \((0, 0)\) and \((0, 4)\) on the \(y\)-axis, and no \(x\)-intercepts exist. The circle should cross the \(y\)-axis at these points.

Key Concepts

Circle Geometry

Circle geometry is all about understanding the properties and equations that define a circle in geometry. A circle is a perfectly round shape where every point on the edge is the same distance from its center. This distance is called the radius. In our exercise, the circle's center, or point \( C \), is located at \( (0, 2) \), with the radius \( a = 2 \).

To find the equation of a circle, you need two things:

• The coordinates of the center \((h, k)\)
• The radius \( a \)

The equation that represents any circle is given by:

• \((x - h)^2 + (y - k)^2 = a^2\)

This equation is derived from the Pythagorean Theorem and ensures that for any point \((x, y)\) on the circle, the distance from \((x, y)\) to the center \((h, k)\) is exactly \( a \).

Understanding this equation is vital because it is the foundation for locating any point on the circumference, and determining intersection points with axes.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where points are defined using coordinates. In the Cartesian plane with an \( x \)-axis and a \( y \)-axis, a circle's location and size can be clearly defined by its equation.

By using the circle equation \( (x - h)^2 + (y - k)^2 = a^2 \), you're actually plotting all points \((x, y)\) that are the radius \( a \) away from the center chosen. In our specific task, the values are fairly straightforward:

• Center \((h, k) = (0, 2)\)
• Radius \( a = 2 \)

This means any point that satisfies the equation \( x^2 + (y - 2)^2 = 4 \) is on the circle. Coordinate geometry helps us move from abstract numbers and letters to tangible, visual points on a graph.

The beauty of coordinate geometry is it allows you to determine intercepts:

• \( y \)-intercepts: set \( x=0 \) and solve: \( (y-2)^2 = 4 \) results in \( y = 0 \) and \( y = 4 \)
• \( x \)-intercepts: set \( y=0 \) and solve: \( x^2 + 4 = 4 \) resulting in no real solutions for \( x \)

This process helps students visualize how circles intersect axes at specific points.

Graphing Equations

Graphing equations lets us visualize the equations we are working with. A graph is essentially a picture of an equation that shows all the possible solutions. For a circle, graphing involves drawing all points equidistant from the center on a coordinate plane.

To graph the equation \( x^2 + (y - 2)^2 = 4 \):

• Plot the center, \((0, 2)\), on the coordinate plane.
• Use the radius \( a = 2 \) to measure out from the center in all directions. This forms a circle.
• Ensure the circle touches the \( y \)-axis exactly at the intercepts \((0, 0)\) and \((0, 4)\).

Graphing helps confirm the situation is fully understood. It visually verifies the algebra by showing the real shape and alignment of the circle as defined by the equation. This process transforms numbers and letters into a more immediate and powerful form. Understanding how to accurately plot a graph from an equation is a crucial skill in both mathematics and various real-world applications.