Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 23-34: Sketch the graph of the circle or semicircle. $$ x^{2}+y^{2}=11 $$

Step-by-Step Solution

Answer

The graph is a circle centered at the origin with radius \(\sqrt{11}\).

Step 1: Identify the Equation Type

The given equation is in the form \(x^2 + y^2 = r^2\), which represents a circle centered at the origin \((0, 0)\) with radius \(r\). In the equation \(x^2 + y^2 = 11\), \(r^2 = 11\).

Step 2: Calculate the Radius

We need to find the radius \(r\) by taking the square root of \(11\). Therefore, \(r = \sqrt{11}\).

Step 3: Sketch the Graph

Draw a coordinate plane with an x-axis and a y-axis intersecting at the origin. Measure out the distance \(\sqrt{11}\) from the origin in all directions (up, down, left, right) on both axes, since the center of the circle is \((0, 0)\). Draw a full circle through these points, maintaining a constant distance of \(\sqrt{11}\) from the center at all points on the circle.

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental tool in graphing and geometry. It consists of two perpendicular lines, which form the x-axis (horizontal line) and y-axis (vertical line). These axes intersect at a point called the origin, designated as \((0, 0)\). Think of the coordinate plane as a map where we can locate points and graph shapes, such as circles and lines.

When graphing on the coordinate plane, it's important to understand the layout:

• The x-axis usually runs left to right.
• The y-axis runs up and down.
• Points are named using ordered pairs \((x, y)\), indicating their distance from the origin.

This layout allows us to precisely describe the position of any point. In the context of a circle, the origin or another specified point on the coordinate plane often serves as the center of the circle.

Circle Equation

The circle equation in the standard form is given by \(x^2 + y^2 = r^2\), where \((x, y)\) represents the coordinates of any point on the circle, and \(r\) is the radius. This equation shows that every point on the circle is at a fixed distance, the radius, from the center point, usually \((0, 0)\) in these simple formulas.

Here's how to interpret the circle equation:

• The expression \(x^2 + y^2\) represents the squared distance from any point \((x,y)\) to the origin.
• The equals sign and \(r^2\) indicates that this distance remains constant for all points on the circle.

Therefore, this circle will equally stretch around the center point in the coordinate plane, forming the circle's boundary. The equation naturally helps in plotting a circle as it dictates all the points that will satisfy the circle's constant radius.

Radius Calculation

The radius of a circle is the distance from the center to any point on the circle's edge. In the equation \(x^2 + y^2 = r^2\), the term \(r^2\) represents the square of the circle's radius. To find the radius itself, we take the square root of \(r^2\).

In the given problem, the circle equation is \(x^2 + y^2 = 11\).
Here, \(r^2 = 11\), which suggests we need to calculate \(r\) by addressing the following:

• Apply the square root: \(r = \sqrt{11}\).
• This value represents the length from the center point of the circle (origin) to any point on its circumference.

Understanding this calculation helps us to draw the circle accurately, be it physically on paper or using a digital tool, ensuring the circle reflects the intended size as per the equation provided.