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`

Options

A. `35 lt m lt 85`
B. `-85 lt m lt - 35`
C. `-35 lt m lt 15`
D. `15 lt m lt 65`

Explanation

To solve the problem, we need to determine 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 circle equation in standard form
The given equation of the circle is:
\[ x^2 + y^2 - 4x - 8y - 5 = 0 \]
We will complete the square for both \( x \) and \( y \).

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

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

Substituting these back into the circle's equation:
\[ (x - 2)^2 - 4 + (y - 4)^2 - 16 - 5 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 4)^2 - 25 = 0 \]
Thus, we have:
\[ (x - 2)^2 + (y - 4)^2 = 25 \]
This represents a circle with center \( (2, 4) \) and radius \( 5 \).

### Step 2: Find the length of the perpendicular from the center to the line
The line equation can be rewritten in the standard form \( Ax + By + C = 0 \):
\[ 3x - 4y - m = 0 \]
Here, \( A = 3 \), \( B = -4 \), and \( C = -m \).

The length \( p \) of the perpendicular from the center \( (2, 4) \) to the line is given by the formula:
\[ p = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \( (x_1, y_1) = (2, 4) \):
\[ p = \frac{|3(2) - 4(4) - m|}{\sqrt{3^2 + (-4)^2}} = \frac{|6 - 16 - m|}{\sqrt{9 + 16}} = \frac{| -10 - m|}{5} \]

### Step 3: Set the condition for intersection at two distinct points
For the line to intersect the circle at two distinct points, the length of the perpendicular \( p \) must be less than the radius of the circle:
\[ \frac{| -10 - m|}{5} < 5 \]
Multiplying both sides by \( 5 \):
\[ | -10 - m| < 25 \]

### Step 4: Solve the inequality
This absolute value inequality can be split into two cases:
1. \( -10 - m < 25 \)
2. \( -10 - m > -25 \)

**Case 1:**
\[ -10 - m < 25 \implies -m < 35 \implies m > -35 \]

**Case 2:**
\[ -10 - m > -25 \implies -m > -15 \implies m < 15 \]

### Final Result
Combining both inequalities, we get:
\[ -35 < m < 15 \]

Thus, the values of \( m \) for which the line intersects the circle at two distinct points are:
\[ \boxed{(-35, 15)} \]