Problem 53 - Algebra and Trigonometry

Question

\(49-54\) . Find the center and radius of the circle, and sketch its graph. $$ (x+3)^{2}+(y-4)^{2}=25 $$

Step-by-Step Solution

Answer

The center is \((-3,4)\) and the radius is \(5\).

Step 1: Identify the Standard Form of the Circle Equation

The given equation \((x+3)^2 + (y-4)^2 = 25\) is already in the standard form of a circle equation, which is \((x-h)^2 + (y-k)^2 = r^2\). From this, we can easily identify the center \((h, k)\) and the radius \(r\) of the circle.

Step 2: Determine the Center of the Circle

By comparing the given equation \((x+3)^2 + (y-4)^2 = 25\) with the standard form \((x-h)^2 + (y-k)^2 = r^2\), we can see that \(h = -3\) and \(k = 4\). Thus, the center of the circle is \((-3, 4)\).

Step 3: Calculate the Radius of the Circle

From the equation \((x+3)^2 + (y-4)^2 = 25\), we see that the right-hand side is \(r^2 = 25\). Taking the square root of both sides, we find that \(r = 5\). Therefore, the radius of the circle is \(5\).

Step 4: Sketch the Graph of the Circle

To sketch the circle, plot the center at \((-3, 4)\) on the coordinate plane. Since the radius is \(5\), draw a circle around this center point with a radius of \(5\) units. Make sure that all points on the circle are \(5\) units away from the center point \((-3, 4)\).

Key Concepts

Center of Circle

To understand the center of a circle from its equation, we need to look at the standard form: \[(x - h)^2 + (y - k)^2 = r^2\]In this equation,

• \(h\) is the x-coordinate of the circle's center.
• \(k\) is the y-coordinate of the center.

Given the example equation \[(x+3)^2+(y-4)^2=25\]we match it with the general formula. By comparing these, we find:

• The expression \((x+3)\) suggests \(h = -3\), because \((x-h)^2\) means \(x\) is being subtracted by \(-3\).
• Similarly, \((y-4)\) makes \(k = 4\). This gives the y-coordinate.

Hence, the center of the circle is at \((-3, 4)\).

Identifying the center is fundamental for locating the circle on a graph because it serves as the fixed point from which all distances (radii) are measured.

Radius of Circle

The radius of a circle is the distance from its center to any point on its circumference. It's crucial for defining the size of the circle. The standard circle equation provides this information through \(r^2\). Hence, the equation is structured as:\[(x-h)^2 + (y-k)^2 = r^2\]From the given problem, the equation is:\[(x+3)^2+(y-4)^2=25\]This can be interpreted as \(r^2 = 25\).To find \(r\), calculate the square root of both sides:

• \(r = \sqrt{25}\)
• \(r = 5\)

The radius of the circle here is \(5\) units. Remember, the radius must always be a positive number and represents the consistent distance each point on the circle stretches from its center.

Graphing Circles

Graphing a circle involves plotting its center on the coordinate plane and using the radius to draw its circumference.**Steps to sketch a circle:**1. **Locate the Center:** - Begin by plotting the center of the circle. For this exercise, the center is \((-3, 4)\). Mark this point on the coordinate graph.2. **Mark Out the Radius:** - From the center, measure a distance equal to the radius (which is 5 units). Move in all direction: up, down, left, and right to mark the extent of the circle.3. **Draw the Circle:** - Using a steady hand or a compass, sketch a round shape that passes through these points. This forms all points equidistant (5 units) from the center.By following these steps, you'll carefully create a circle that accurately represents \((x+3)^2 + (y-4)^2 = 25\). Each point of the circle is always exactly 5 units away from the center at \((-3, 4)\). Understanding how to graph a circle is essential, as it visually represents algebraic definitions in geometry.