**Option A.** 2 units

**Option B.** 3 units

**Option C.** 4 units

**Option D.** 5 units

**Ans:** The radius ‘r’ of the circle with the equation x² + y² + 8x – 6y + 21 = 0 is 5 units.

## Explanation: for What is the radius of a circle whose equation is x2+y2+8x−6y+21=0? 2 units 3 units 4 units 5 units

In the realm of mathematics, circles are a well-studied geometric shape, often described by equations that encapsulate their unique properties. When presented with the equation x² + y² + 8x – 6y + 21 = 0, the task at hand is to unveil the radius of the corresponding circle. To achieve this, we’ll employ the standard circle equation and follow a systematic process.

## The General Form of a Circle Equation

The general equation representing a circle in Cartesian coordinates is as follows:

(x – a)² + (y – b)² = r²

Here, (a, b) represents the center of the circle, and r is the radius. Our objective is to find the values of a, b, and ultimately, r, from the given equation.

## Step 1: Comparison with the General Form

To begin, we compare the provided equation, x² + y² + 8x – 6y + 21 = 0, with the general form:

(x – a)² + (y – b)² = r²

By making this comparison, we can discern the values of a and b.

## Step 2: Finding ‘a’

For the x-component of the equation, we have:

-2a = 8x

Solving for ‘a,’ we find:

a = -4

## Step 3: Finding ‘b’

Similarly, for the y-component of the equation, we have:

-2b = -6y

Solving for ‘b,’ we find:

b = 3

## Step 4: Calculating the Radius ‘r’

Now that we have determined the values of ‘a’ and ‘b,’ we can calculate the radius ‘r’ using the formula:

r = √(a² + b²)

Plugging in the values we found:

r = √((-4)² + 3²) = √(16 + 9) = √25 = 5

## The Result

Hence, the radius ‘r’ of the circle with the equation x² + y² + 8x – 6y + 21 = 0 is 5 units. This result confirms that the circle is centered at (-4, 3) and has a radius of 5 units. The ability to extract such information from equations is a fundamental skill in algebra and geometry, with practical applications across various mathematical and scientific disciplines.

This solution aligns with the conclusion reached earlier and demonstrates the consistency of mathematical methodologies. The process of identifying key parameters from equations is a valuable tool in problem-solving and mathematical analysis.