If the circle `x^(2)+y^(2)=4x+8y+5` intersects the line `3x-4y=m` at two distinct points, then the number of possible integral values of m is equal to

Options

A. 51
B. 50
C. 49
D. 48

Answer

To solve the problem, we need to determine the number of integral values of \( m \) for which the line \( 3x - 4y = m \) intersects the circle defined by the equation \( x^2 + y^2 = 4x + 8y + 5 \) at two distinct points.

### Step 1: Rewrite the circle equation
The given circle equation is:
\[ x^2 + y^2 = 4x + 8y + 5 \]
Rearranging this, we get:
\[ x^2 - 4x + y^2 - 8y - 5 = 0 \]

### Step 2: Complete the square
To convert this into standard form, we complete the square for \( x \) and \( y \).

For \( x^2 - 4x \):
\[ x^2 - 4x = (x - 2)^2 - 4 \]

For \( y^2 - 8y \):
\[ y^2 - 8y = (y - 4)^2 - 16 \]

Substituting these back into the equation gives:
\[ (x - 2)^2 - 4 + (y - 4)^2 - 16 - 5 = 0 \]
\[ (x - 2)^2 + (y - 4)^2 - 25 = 0 \]
\[ (x - 2)^2 + (y - 4)^2 = 25 \]

### Step 3: Identify the center and radius
From the standard form, we can see that the center of the circle is \( (2, 4) \) and the radius \( r = 5 \).

### Step 4: Find the distance from the center to the line
The line equation is given as \( 3x - 4y = m \). The distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For our line \( 3x - 4y - m = 0 \), we have \( A = 3 \), \( B = -4 \), and \( C = -m \). The center of the circle \( (2, 4) \) gives us:
\[ d = \frac{|3(2) - 4(4) - m|}{\sqrt{3^2 + (-4)^2}} = \frac{|6 - 16 - m|}{\sqrt{9 + 16}} = \frac{| -10 - m|}{5} \]

### Step 5: Set up the inequality for intersection
For the line to intersect the circle at two distinct points, the distance from the center to the line must be less than the radius:
\[ \frac{| -10 - m|}{5} < 5 \]
Multiplying both sides by 5 gives:
\[ | -10 - m| < 25 \]

### Step 6: Solve the absolute value inequality
This leads to two inequalities:
\[ -25 < -10 - m < 25 \]

1. From \( -25 < -10 - m \):
\[ -25 + 10 < -m \implies -15 < -m \implies m < 15 \]

2. From \( -10 - m < 25 \):
\[ -10 - m < 25 \implies -m < 35 \implies m > -35 \]

Combining these results, we have:
\[ -35 < m < 15 \]

### Step 7: Count the integral values
The integral values of \( m \) in the range \( -34 \) to \( 14 \) (inclusive) can be calculated as follows:
- The integers from \( -34 \) to \( 14 \) are:
\[ -34, -33, -32, \ldots, 0, 1, 2, \ldots, 14 \]
- The total number of integers from \( -34 \) to \( 14 \) is:
\[ 14 - (-34) + 1 = 14 + 34 + 1 = 49 \]

### Final Answer
Thus, the number of possible integral values of \( m \) is \( \boxed{49} \).