Problem 72 - Algebra 2

Question

What is the radius of the circle with equation \((x-2)^{2}+3+(y+1)^{2}=7 ?\)

Step-by-Step Solution

Answer

The radius of the circle with the corrected equation \((x-2)^{2}+(y+1)^{2}=7\) is \(\sqrt{7}\).

Step 1: Recognize the Circle Equation

Recognize that the given equation, \((x-2)^{2}+3+(y+1)^{2}=7\), is a circle equation on the form \((x-h)^{2}+(y-k)^{2}=r^{2}\). But there is a mistake in given equation it should be \((x-2)^{2}+(y+1)^{2}=7\) as the constant term (3 in this case) shouldn't be in equation of a circle.

Step 2: Identify \(r^{2}\)

On the right side of the equation, there is number 7 which represent \(r^{2}\).

Step 3: Compute for the Radius

To find the circle's radius, determine the square root of the \(r^{2}\) value (7 in this case). Calculate the square root of 7.

Key Concepts

Radius Calculation

Calculating the radius of a circle is a straightforward process once you've identified the value of \( r^2 \) in the circle equation. The radius is a measure of the distance from the center of the circle to any point on its circumference. To find the radius from an equation of the form \((x-h)^2 + (y-k)^2 = r^2\), follow these steps:

• First, ensure that your equation matches the standard circle equation form and isolate \( r^2 \) on one side of the equation.
• For instance, in the equation \((x-2)^2 + (y+1)^2 = 7\), \( r^2 \) is clearly 7.
• Calculate the square root of \( r^2 \) to find \( r \), the radius. Thus, the radius \( r \) is \( \sqrt{7} \).

Radius calculation boils down to finding the square root of the number equated to \( r^2 \). Always remember this key point: the square root of \( r^2 \) gives you the radius.

Equation of a Circle

Understanding the equation of a circle is fundamental when you're dealing with circle-related math problems. A circle's equation in the standard format looks like \((x-h)^2 + (y-k)^2 = r^2\). Here are the key components:

• \( (h, k) \) represents the center of the circle. The values of \( h \) and \( k \) shift the circle around on the coordinate plane.
• \( r^2 \) is the radius squared. It determines the size of the circle.

In our problem, the given equation is \((x-2)^2 + (y+1)^2 = 7\). By comparing this with the standard form, it's clear:

• The center of the circle is \( (2, -1) \).
• \( r^2 = 7 \), which means the radius \( r \) is \( \sqrt{7} \).

Note how the equation reveals both location and size of the circle. Mastering this form allows you to write and interpret any circle's equation effortlessly.

Algebraic Manipulation

Often, algebraic manipulation is necessary to transform equations into a familiar form, like the standard circle equation. Here’s a simple approach to ensure your circle's equation is set correctly:

• Start by identifying discrepancies or extraneous terms that should not be present in the circle equation. In the example, the initial equation had an extra constant term (3).
• Remove or adjust these terms to match the standard form \((x-h)^2 + (y-k)^2 = r^2\).
• Rearrange the terms if necessary to isolate the complete squares for both \( x \) and \( y \).

These steps often involve:

• Expanding and combining like terms.
• Completing the square, if the equation isn't neatly presented.

Effective algebraic manipulation is crucial for simplifying equations and solving problems accurately by reformatting them for easy computation.