The line `3x -4y = k` will cut the circle `x^(2) + y^(2) -4x -8y -5 = 0` at distinct points if

Options

A. `-10ltlamdalt5`
B. `9ltlamdalt20`
C. `-35ltlamdalt15`
D. `-16ltlamdalt30`

Answer

To determine the values of \( k \) for which the line \( 3x - 4y = k \) intersects the circle given by the equation \( x^2 + y^2 - 4x - 8y - 5 = 0 \) at distinct points, we can follow these steps:

### Step 1: Rewrite the Circle Equation
First, we rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 - 4x - 8y - 5 = 0 \]
We can complete the square for \( x \) and \( y \).

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

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

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

This shows that the circle has a center at \( (2, 4) \) and a radius of \( 5 \).

### Step 2: Find the Distance from the Center to the Line
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 the line \( 3x - 4y = k \), we can rewrite it as \( 3x - 4y - k = 0 \). Here, \( A = 3 \), \( B = -4 \), and \( C = -k \).

Substituting the center of the circle \( (2, 4) \):
\[ d = \frac{|3(2) - 4(4) - k|}{\sqrt{3^2 + (-4)^2}} = \frac{|6 - 16 - k|}{\sqrt{9 + 16}} = \frac{| -10 - k |}{5} \]

### Step 3: Set Up the Inequality for Distinct Points
For the line to intersect the circle at distinct points, the distance from the center to the line must be less than the radius:
\[ \frac{| -10 - k |}{5} < 5 \]
Multiplying both sides by \( 5 \):
\[ | -10 - k | < 25 \]

### Step 4: Solve the Inequality
This absolute value inequality can be split into two inequalities:
\[ -25 < -10 - k < 25 \]

Rearranging the left part:
\[ -25 + 10 < -k \implies -15 < -k \implies k < 15 \]

Rearranging the right part:
\[ -10 - k < 25 \implies -k < 35 \implies k > -35 \]

Combining these results gives:
\[ -35 < k < 15 \]

### Conclusion
Thus, the line \( 3x - 4y = k \) will cut the circle at distinct points if:
\[ -35 < k < 15 \]