Problem 54 - Algebra and Trigonometry

Question

\(49-54\) . Find the center and radius of the circle, and sketch its graph. $$ (x+1)^{2}+(y+2)^{2}=36 $$

Step-by-Step Solution

Answer

Center: (-1, -2); Radius: 6.

Step 1: Identify the Circle Equation Form

Recognize the given circle equation \[(x+1)^2 + (y+2)^2 = 36\]as being in the standard form for 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 the Center

Compare \((x+1)^2 + (y+2)^2 = 36\)to \((x-h)^2 + (y-k)^2 = r^2\). The center is at \((h, k)\).Thus, \(h = -1\) and \(k = -2\).So, the center of the circle is \((-1, -2)\).

Step 3: Calculate the Radius

The right-hand side of the equation, 36, is \(r^2\), so find \(r\)by taking the square root of 36.\[r = \sqrt{36} = 6\]Thus, the radius of the circle is 6.

Step 4: Sketch the Circle

To sketch the circle, first plot the center at \((-1, -2)\).From this point, measure out the radius of 6 units in all directions (up, down, left, right) to outline the circle. Ensure that the circle is round and passes through each of these points to accurately represent the radius.

Key Concepts

Center of a Circle

The center of a circle is a crucial point within the circle, from which all points on the circle are equidistant. It acts as the fixed position, anchoring the circle in the plane.
In a circle's equation written in the standard form, o\[(x-h)^2 + (y-k)^2 = r^2\]the coordinates of the center are o\((h, k)\).When identifying the center from an equation, observe the transformations applied to the variables within the square terms.

• The term \(x - h\) inside the squared format implies shifting the circle from the origin by \(h\) units on the x-axis.
• Similarly, \(y - k\) indicates a shift by \(k\) units on the y-axis.

Hence, for the equation o\((x+1)^{2}+(y+2)^{2}=36\), the expressions o\((x+1)\) and o\((y+2)\) can be rewritten as o\((x - (-1))\) and o\((y - (-2))\), effectively indicating the center as o\((-1, -2)\).
Understanding the placement of the center helps in visualizing the circle's position in coordination with its radius.

Radius of a Circle

The radius of a circle is the distance from its center to any point on the circle's boundary. This distance is constant for any circle, making it a fundamental measure that defines the circle's size.
In the standard form equation of a circle, o\[(x-h)^2 + (y-k)^2 = r^2\],the term o\(r^2\)represents the square of the radius.To find the actual radius, simply take the square root of this value:

• If \(r^2 = 36\), taking the square root gives o\(r = \sqrt{36} = 6\).
• Thus, the radius of the circle is 6 units.

Finding the radius allows you to understand how large the circle is and aids in creating an accurate sketch when visualizing or graphing the circle.
It is like measuring the length of a straight line starting from the center and extending to any place along the circumference.

Standard Form of a Circle Equation

The standard form of a circle's equation provides a direct yet comprehensive way to describe any circle on a coordinate plane. This form is denoted as:o\[(x-h)^2 + (y-k)^2 = r^2\],where each element has a specific purpose:

• The components \(x-h\) and \(y-k\) illustrate the horizontal and vertical shifts of the circle from the origin, thereby determining the center location \(h, k\).
• The \(r^2\) term captures the radius squared, simplifying the determination of the circle's size by recognizing it as the distance squared from any boundary point to the center.

Whenever you're presented with an equation that's structured this way, you can easily extract the circle's center and radius. This is achievable by focusing on the termso\((h, k)\) and o\(r^2\).By thoroughly understanding this standard form, you gain the ability to swiftly analyze and interpret any circle equation, making connections between algebraic expressions and geometric concepts.
This form simplifies graphing and ensures accurate plotting of any given circle on a coordinate grid.