If (2, 4) is a point interior to the circle `x^(2)+y^(2)-6x-10y+lambda=0` and the circle does not cut the axes at any point, then

Options

A. `lambda in(25, 32)`
B. `lambda in(9, 32)`
C. `lambda in(32, oo)`
D. none of these

Explanation

To solve the problem, we need to analyze the conditions given in the question step by step.

### Step 1: Understand the Circle Equation
The equation of the circle is given as:
\[ x^2 + y^2 - 6x - 10y + \lambda = 0 \]
We can rewrite this in the standard form of a circle by completing the square.

### Step 2: Complete the Square
1. For \(x^2 - 6x\):
\[ x^2 - 6x = (x - 3)^2 - 9 \]
2. For \(y^2 - 10y\):
\[ y^2 - 10y = (y - 5)^2 - 25 \]

Now substituting these back into the circle equation:
\[ (x - 3)^2 - 9 + (y - 5)^2 - 25 + \lambda = 0 \]
This simplifies to:
\[ (x - 3)^2 + (y - 5)^2 + \lambda - 34 = 0 \]
Thus, the standard form of the circle is:
\[ (x - 3)^2 + (y - 5)^2 = 34 - \lambda \]

### Step 3: Condition for Not Cutting the Axes
For the circle not to cut the axes, the radius must be less than the distance from the center to the axes. The center of the circle is \((3, 5)\).

1. Distance from center to the x-axis is \(5\).
2. Distance from center to the y-axis is \(3\).

The radius \(r\) of the circle is given by:
\[ r = \sqrt{34 - \lambda} \]
For the circle not to intersect the axes:
\[ \sqrt{34 - \lambda} < 3 \quad \text{and} \quad \sqrt{34 - \lambda} < 5 \]

### Step 4: Solve the Inequalities
1. From \(\sqrt{34 - \lambda} < 3\):
\[ 34 - \lambda < 9 \implies \lambda > 25 \]
2. From \(\sqrt{34 - \lambda} < 5\):
\[ 34 - \lambda < 25 \implies \lambda > 9 \]
(This condition is always satisfied since \( \lambda > 25 \) is stricter.)

### Step 5: Condition for the Point (2, 4) to be Interior
For the point \((2, 4)\) to be inside the circle:
\[ (2 - 3)^2 + (4 - 5)^2 < 34 - \lambda \]
Calculating the left side:
\[ 1 + 1 < 34 - \lambda \implies 2 < 34 - \lambda \implies \lambda < 32 \]

### Step 6: Combine Conditions
From the two conditions:
1. \(\lambda > 25\)
2. \(\lambda < 32\)

Thus, we conclude:
\[ 25 < \lambda < 32 \]

### Final Answer
The value of \(\lambda\) belongs to the interval:
\[ \lambda \in (25, 32) \]