If the points `(lambda, -lambda)` lies inside the circle `x^2 + y^2 - 4x + 2y -8=0`, then find the range of `lambda`.

Answer

To solve the problem of finding the range of \( \lambda \) such that the point \( (\lambda, -\lambda) \) lies inside the circle defined by the equation \( x^2 + y^2 - 4x + 2y - 8 = 0 \), we will follow these steps:

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

### Step 2: Complete the Square
For \( x \):
\[ x^2 - 4x \quad \text{can be rewritten as} \quad (x - 2)^2 - 4 \]
For \( y \):
\[ y^2 + 2y \quad \text{can be rewritten as} \quad (y + 1)^2 - 1 \]
Now substituting these back into the equation:
\[ (x - 2)^2 - 4 + (y + 1)^2 - 1 - 8 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y + 1)^2 - 13 = 0 \]
Thus, we have:
\[ (x - 2)^2 + (y + 1)^2 = 13 \]
This represents a circle with center \( (2, -1) \) and radius \( \sqrt{13} \).

### Step 3: Substitute the Point into the Circle Equation
We need to check when the point \( (\lambda, -\lambda) \) lies inside the circle. We substitute \( x = \lambda \) and \( y = -\lambda \) into the circle equation:
\[ (\lambda - 2)^2 + (-\lambda + 1)^2 < 13 \]

### Step 4: Expand and Simplify the Inequality
Expanding the left-hand side:
\[ (\lambda - 2)^2 = \lambda^2 - 4\lambda + 4 \]
\[ (-\lambda + 1)^2 = \lambda^2 - 2\lambda + 1 \]
Adding these gives:
\[ \lambda^2 - 4\lambda + 4 + \lambda^2 - 2\lambda + 1 < 13 \]
This simplifies to:
\[ 2\lambda^2 - 6\lambda + 5 < 13 \]
Subtracting 13 from both sides:
\[ 2\lambda^2 - 6\lambda - 8 < 0 \]

### Step 5: Factor the Quadratic Inequality
Dividing the entire inequality by 2:
\[ \lambda^2 - 3\lambda - 4 < 0 \]
Now we factor the quadratic:
\[ (\lambda - 4)(\lambda + 1) < 0 \]

### Step 6: Determine the Range of \( \lambda \)
To find the intervals where this product is negative, we analyze the critical points \( \lambda = -1 \) and \( \lambda = 4 \). The intervals to test are:
1. \( \lambda < -1 \)
2. \( -1 < \lambda < 4 \)
3. \( \lambda > 4 \)

Testing these intervals:
- For \( \lambda < -1 \), both factors are negative, so the product is positive.
- For \( -1 < \lambda < 4 \), one factor is negative and the other is positive, so the product is negative.
- For \( \lambda > 4 \), both factors are positive, so the product is positive.

Thus, the solution to the inequality is:
\[ \lambda \in (-1, 4) \]

### Final Answer
The range of \( \lambda \) such that the point \( (\lambda, -\lambda) \) lies inside the circle is:
\[ \boxed{(-1, 4)} \]