If the line `3x-4y=lamda` cuts the circle `x^(2)+y^(2)-4x-8y-5=0` in two points then limits of `lamda` are

Options

A. `[-35,15]`
B. `(-35,15)`
C. `(-35,10)`
D. `(-35,15]`

Answer

To find the limits of \(\lambda\) such that the line \(3x - 4y = \lambda\) intersects the circle defined by the equation \(x^2 + y^2 - 4x - 8y - 5 = 0\) at two 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 gives:
\[ (x - 2)^2 - 4 + (y - 4)^2 - 16 - 5 = 0 \]
\[ (x - 2)^2 + (y - 4)^2 - 25 = 0 \]
\[ (x - 2)^2 + (y - 4)^2 = 25 \]

This shows that the center of the circle is \((2, 4)\) and the radius \(r\) is \(5\).

### Step 2: Find the Perpendicular Distance from the Center to the Line
The distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

For the line \(3x - 4y = \lambda\), we can rewrite it as:
\[ 3x - 4y - \lambda = 0 \]
Here, \(A = 3\), \(B = -4\), and \(C = -\lambda\). The center of the circle is \((2, 4)\).

Calculating the distance:
\[ d = \frac{|3(2) - 4(4) - \lambda|}{\sqrt{3^2 + (-4)^2}} = \frac{|6 - 16 - \lambda|}{\sqrt{9 + 16}} = \frac{| -10 - \lambda|}{5} \]

### Step 3: Set Up the Inequality for Intersection
For the line to intersect the circle at two points, the distance \(d\) must be less than the radius:
\[ \frac{| -10 - \lambda|}{5} < 5 \]

Multiplying both sides by \(5\):
\[ | -10 - \lambda| < 25 \]

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

Breaking it down:
1. From \(-10 - \lambda < 25\):
\[ -\lambda < 35 \implies \lambda > -35 \]

2. From \(-25 < -10 - \lambda\):
\[ -25 + 10 < -\lambda \implies -15 < -\lambda \implies \lambda < 15 \]

### Step 5: Combine the Results
Combining the results from the inequalities, we have:
\[ -35 < \lambda < 15 \]

### Conclusion
Thus, the limits of \(\lambda\) such that the line intersects the circle at two points are:
\[ \lambda \in (-35, 15) \]