The y-intercept of the circle `x^(2)+y^(2)+4x+8y-5=0` is

Options

A. `2sqrt(21)`
B. `2sqrt(19)`
C. 6
D. 12

Answer

To find the y-intercept of the circle given by the equation \(x^2 + y^2 + 4x + 8y - 5 = 0\), we will follow these steps:

### Step 1: Substitute \(x = 0\)
To find the y-intercept, we set \(x = 0\) in the equation of the circle. This is because the y-intercept occurs where the circle intersects the y-axis, which is defined by \(x = 0\).

\[ 0^2 + y^2 + 4(0) + 8y - 5 = 0 \]
This simplifies to:
\[ y^2 + 8y - 5 = 0 \]

### Step 2: Rearrange the equation
Now, we rearrange the equation to make it easier to solve:
\[ y^2 + 8y - 5 = 0 \]

### Step 3: Use the quadratic formula
To solve for \(y\), we can use the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 8\), and \(c = -5\).

### Step 4: Calculate the discriminant
First, we calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = 8^2 - 4(1)(-5) = 64 + 20 = 84 \]

### Step 5: Substitute values into the quadratic formula
Now we substitute \(a\), \(b\), and the discriminant into the quadratic formula:
\[ y = \frac{-8 \pm \sqrt{84}}{2(1)} = \frac{-8 \pm \sqrt{84}}{2} \]

### Step 6: Simplify the expression
We can simplify \(\sqrt{84}\):
\[ \sqrt{84} = \sqrt{4 \times 21} = 2\sqrt{21} \]
So now we have:
\[ y = \frac{-8 \pm 2\sqrt{21}}{2} \]
This simplifies to:
\[ y = -4 \pm \sqrt{21} \]

### Step 7: Identify the y-intercepts
Thus, the two y-intercepts are:
1. \(y = -4 + \sqrt{21}\)
2. \(y = -4 - \sqrt{21}\)

### Step 8: Calculate the distance between the two y-intercepts
The distance between these two points on the y-axis gives us the y-intercept:
\[ \text{Distance} = (-4 + \sqrt{21}) - (-4 - \sqrt{21}) = \sqrt{21} + \sqrt{21} = 2\sqrt{21} \]

### Final Answer
Thus, the y-intercept of the circle is:
\[ \boxed{2\sqrt{21}} \]