If P(2,8) is an interior point of a circle `x^(2)+y^(2)-2x+4y-lamda=0` which neither touches nor intersects the axes, then set for `lamda` is

Options

A. `(-oo,-1)`
B. `(-oo,-4)`
C. `(96,oo)`
D. `phi`

Answer

To solve the problem, we need to determine the set of values for \(\lambda\) such that the point \(P(2, 8)\) is an interior point of the circle defined by the equation \(x^2 + y^2 - 2x + 4y - \lambda = 0\), and the circle neither touches nor intersects the axes.

### Step 1: Identify the general form of the circle
The equation of the circle can be rewritten in the standard form:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \(g = -1\), \(f = 2\), and \(c = -\lambda\).

### Step 2: Determine the condition for the point \(P(2, 8)\) to be inside the circle
For a point \((x_1, y_1)\) to be inside the circle, we need to evaluate:
\[ S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c < 0 \]
Substituting \(P(2, 8)\) into the equation:
\[ S_1 = 2^2 + 8^2 + 2(-1)(2) + 2(2)(8) - \lambda \]
Calculating each term:
\[ S_1 = 4 + 64 - 4 + 32 - \lambda \]
Simplifying:
\[ S_1 = 96 - \lambda < 0 \]
This leads to:
\[ \lambda > 96 \quad \text{(Condition 1)} \]

### Step 3: Determine the conditions for the circle not to touch or intersect the axes
1. **Condition for the x-axis**: The circle does not touch or intersect the x-axis if:
\[ g^2 - c < 0 \]
Substituting \(g = -1\) and \(c = -\lambda\):
\[ (-1)^2 - (-\lambda) < 0 \]
This simplifies to:
\[ 1 + \lambda < 0 \implies \lambda < -1 \quad \text{(Condition 2)} \]

2. **Condition for the y-axis**: The circle does not touch or intersect the y-axis if:
\[ f^2 - c < 0 \]
Substituting \(f = 2\) and \(c = -\lambda\):
\[ (2)^2 - (-\lambda) < 0 \]
This simplifies to:
\[ 4 + \lambda < 0 \implies \lambda < -4 \quad \text{(Condition 3)} \]

### Step 4: Analyze the conditions
We have three conditions:
1. \(\lambda > 96\)
2. \(\lambda < -1\)
3. \(\lambda < -4\)

The first condition states that \(\lambda\) must be greater than 96, while the other two conditions require \(\lambda\) to be less than -1 and -4, respectively.

### Conclusion
There is no value of \(\lambda\) that can satisfy all three conditions simultaneously. Therefore, the answer is that there does not exist any value of \(\lambda\) that meets the criteria.

### Final Answer
The set for \(\lambda\) is empty, indicating that no such \(\lambda\) exists.

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