The intercept made by the circle `x^(2)+y^(2)+4x-8y+c=0` on x-axis is `2sqrt(10)` then c=

Options

A. `-6`
B. `6`
C. `+-6`
D. `12`

Answer

To solve the problem, we need to find the value of \( c \) in the circle equation \( x^2 + y^2 + 4x - 8y + c = 0 \) given that the intercept made by the circle on the x-axis is \( 2\sqrt{10} \).

### Step-by-step solution:

1. **Identify the center of the circle**:
The general form of a circle is given by \( (x - h)^2 + (y - k)^2 = r^2 \). We can rewrite the given equation in standard form by completing the square.
- The equation is \( x^2 + 4x + y^2 - 8y + c = 0 \).
- Completing the square for \( x \):
\[ x^2 + 4x = (x + 2)^2 - 4 \]
- Completing the square for \( y \):
\[ y^2 - 8y = (y - 4)^2 - 16 \]
- Substituting back, we get:
\[ (x + 2)^2 - 4 + (y - 4)^2 - 16 + c = 0 \]
- This simplifies to:
\[ (x + 2)^2 + (y - 4)^2 + (c - 20) = 0 \]
- Thus, the center of the circle is \( (-2, 4) \).

2. **Determine the radius**:
The radius \( r \) can be found using the intercept on the x-axis. The total intercept on the x-axis is given as \( 2\sqrt{10} \), which means the distance from the center to the x-axis is \( 4 \) (the y-coordinate of the center) and the half-length of the intercept is \( \sqrt{10} \).
- Therefore, the radius \( r \) can be calculated using the Pythagorean theorem:
\[ r^2 = (4)^2 + (\sqrt{10})^2 \]
\[ r^2 = 16 + 10 = 26 \]
- Hence, \( r = \sqrt{26} \).

3. **Relate the radius to the equation**:
From the standard form of the circle, we know that:
\[ r^2 = g^2 + f^2 - c \]
where \( g = -2 \) and \( f = 4 \).
- Thus, we can substitute:
\[ r^2 = (-2)^2 + (4)^2 - c \]
\[ 26 = 4 + 16 - c \]
\[ 26 = 20 - c \]

4. **Solve for \( c \)**:
Rearranging the equation gives:
\[ c = 20 - 26 \]
\[ c = -6 \]

### Final Answer:
The value of \( c \) is \( -6 \).