Problem 8 - Algebra 2

Question

Find the center and radius of the circle with the given equation. Then graph the circle. $$ x^{2}+(y-14)^{2}=34 $$

Step-by-Step Solution

Answer

Center: (0, 14); Radius: \(\sqrt{34}\).

Step 1: Identify Standard Form of a Circle Equation

The standard equation of a circle in the Cartesian coordinate plane is given by \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius of the circle. Compare the given equation \(x^2 + (y - 14)^2 = 34\) with the standard form.

Step 2: Extract the Circle's Center

From the equation \(x^2 + (y - 14)^2 = 34\), compare it with the standard form \((x-h)^2+(y-k)^2=r^2\). It is apparent that \(h = 0\) and \(k = 14\). Therefore, the center of the circle is \((0, 14)\).

Step 3: Calculate the Circle’s Radius

The given equation is \(x^2 + (y - 14)^2 = 34\). In the standard form, \(r^2 = 34\). Solve for \(r\) by taking the square root of both sides: \(r = \sqrt{34}\).

Step 4: Graph the Circle

Plot the center of the circle on the coordinate plane at \((0, 14)\). From the center, draw a circle with a radius of \(\sqrt{34}\) units. Ensure the drawn circle is equidistant from the center in every direction.

Key Concepts

Standard Form of a Circle

Understanding the standard form of a circle is crucial for solving problems involving circles in mathematics. The equation is written as \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h\) and \(k\) represent the coordinates of the circle's center, and \(r\) stands for the radius.

• Understanding the Equation: This equation helps pinpoint the exact location and size of a circle in the Cartesian plane.
• Importance: Mastery of this form allows one to easily extract the circle's center and radius, crucial for graphing.

Once you recognize a circle's equation is in standard form, comparing it directly to \((x-h)^2 + (y-k)^2 = r^2\) offers a straightforward way to identify its key features.

Center of a Circle

The center of a circle is the point from which every point on the circle is equally distant. In the standard equation, \((h, k)\) are the coordinates of the center.For example, in the equation \(x^2 + (y - 14)^2 = 34\), we can see by comparing it to \((x-h)^2 + (y-k)^2 = r^2\) that:

• \(h = 0\)
• \(k = 14\)

Thus, the center is \((0, 14)\). Knowing the center helps in determining the circle's position on the graph. It's an anchor point from which the circle stretches outward.

Radius of a Circle

The radius of a circle is a measurement from the center to any point on the circle itself. It defines the circle's size. In the standard equation \((x-h)^2 + (y-k)^2 = r^2\), \(r\) is the radius of the circle.From the equation \(x^2 + (y - 14)^2 = 34\), we identify:

• \(r^2 = 34\)

To find \(r\), take the square root of both sides: \(r = \sqrt{34}\). While \(\sqrt{34}\) is an irrational number, approximations can be useful for graphing and practical measurements. It's essential to calculate the radius accurately to draw the circle with the correct size.

Graphing Circles

Graphing circles can be intuitive once you have the center and radius. Start by plotting the center on the coordinate plane.Here are a few steps to guide you:

• Plot the Center: For equation \(x^2 + (y-14)^2 = 34\), plot \((0, 14)\) on the graph.
• Use the Radius: With \(r = \sqrt{34}\), estimate its value for easier plotting.
• Draw the Circle: Placing your compass tip on the center, draw a circle while maintaining the radius length consistently from the center.

The circle should be equidistant from its center in all directions. Drawing it accurately helps visualize important concepts like symmetry and spatial relationships.