If the circle `x^2+y^2-4x-8y-5=0` intersects the line `3x-4y=m` at two distinct points, then find the values of `mdot`

Explanation

To solve the problem, we need to find the values of \( m \) such that the line \( 3x - 4y = m \) intersects the circle given by the equation \( x^2 + y^2 - 4x - 8y - 5 = 0 \) at two distinct points.

### Step 1: Rewrite the equation of the circle
The given equation of the circle is:
\[ x^2 + y^2 - 4x - 8y - 5 = 0 \]
We can rewrite it in standard form by completing the square.

### Step 2: Complete the square
1. For \( x^2 - 4x \):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
2. For \( y^2 - 8y \):
\[ y^2 - 8y = (y - 4)^2 - 16 \]

Now, substituting back into the equation:
\[ (x - 2)^2 - 4 + (y - 4)^2 - 16 - 5 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 4)^2 - 25 = 0 \]
Thus, the equation of the circle in standard form is:
\[ (x - 2)^2 + (y - 4)^2 = 25 \]
This indicates that the center of the circle is \( (2, 4) \) and the radius \( r = 5 \).

### Step 3: Find the distance from the center to the line
The line can be rewritten as:
\[ 3x - 4y - m = 0 \]
To find the distance \( d \) from the center \( (2, 4) \) to the line, we use the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
where \( A = 3 \), \( B = -4 \), \( C = -m \), and \( (x_1, y_1) = (2, 4) \).

Substituting these values:
\[ d = \frac{|3(2) - 4(4) - m|}{\sqrt{3^2 + (-4)^2}} = \frac{|6 - 16 - m|}{\sqrt{9 + 16}} = \frac{| -10 - m|}{5} \]

### Step 4: Set the condition for two distinct intersections
For the line to intersect the circle at two distinct points, the distance \( d \) must be less than the radius \( r \):
\[ \frac{| -10 - m|}{5} < 5 \]
Multiplying both sides by 5:
\[ | -10 - m| < 25 \]

### Step 5: Solve the inequality
This absolute value inequality can be split into two inequalities:
1. \( -10 - m < 25 \)
2. \( -10 - m > -25 \)

Solving the first inequality:
\[ -10 - m < 25 \implies -m < 35 \implies m > -35 \]

Solving the second inequality:
\[ -10 - m > -25 \implies -m > -15 \implies m < 15 \]

### Step 6: Combine the results
Thus, the values of \( m \) for which the line intersects the circle at two distinct points are:
\[ -35 < m < 15 \]

### Final Answer
The values of \( m \) are:
\[ m \in (-35, 15) \]