The radius of circle `x^2+y^2-6x+4y-12=0` is

Options

A. 1
B. 2
C. 3
D. 5

Answer

To find the radius of the circle given by the equation \(x^2 + y^2 - 6x + 4y - 12 = 0\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
The standard form of a circle's equation is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From the given equation, we can identify the coefficients:
- \(2g = -6\) → \(g = -3\)
- \(2f = 4\) → \(f = 2\)
- \(c = -12\)

### Step 2: Use the formula for the radius
The radius \(R\) of the circle can be calculated using the formula:
\[ R = \sqrt{g^2 + f^2 - c} \]

### Step 3: Substitute the values of \(g\), \(f\), and \(c\)
Now we substitute the values we found:
- \(g = -3\)
- \(f = 2\)
- \(c = -12\)

The formula becomes:
\[ R = \sqrt{(-3)^2 + (2)^2 - (-12)} \]

### Step 4: Calculate the values
Calculating each term:
- \((-3)^2 = 9\)
- \((2)^2 = 4\)
- \(-(-12) = 12\)

Now substitute these values back into the equation:
\[ R = \sqrt{9 + 4 + 12} \]

### Step 5: Simplify the expression
Now add the values inside the square root:
\[ R = \sqrt{9 + 4 + 12} = \sqrt{25} \]

### Step 6: Find the final value of the radius
Calculating the square root gives:
\[ R = 5 \]

Thus, the radius of the circle is \(5\).

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