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 given circle equation and the conditions provided. The equation of the circle is:

\[ x^2 + y^2 - 6x - 10y + \lambda = 0 \]

### Step 1: Identify the center and radius of the circle
The general form of a circle is given by:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

We can rewrite the given equation in standard form by completing the square.

1. Rearranging the equation:
\[ x^2 - 6x + y^2 - 10y + \lambda = 0 \]

2. Completing the square for \(x\) and \(y\):
- For \(x^2 - 6x\):
\[ x^2 - 6x = (x - 3)^2 - 9 \]
- For \(y^2 - 10y\):
\[ y^2 - 10y = (y - 5)^2 - 25 \]

3. Substitute back into the equation:
\[ (x - 3)^2 - 9 + (y - 5)^2 - 25 + \lambda = 0 \]
\[ (x - 3)^2 + (y - 5)^2 + \lambda - 34 = 0 \]
\[ (x - 3)^2 + (y - 5)^2 = 34 - \lambda \]

From this, we can see that the center of the circle is \((3, 5)\) and the radius \(r\) is \(\sqrt{34 - \lambda}\).

### Step 2: Condition for the point (2, 4) to be inside the circle
For the point \((2, 4)\) to be inside the circle, the following inequality must hold:

\[ (2 - 3)^2 + (4 - 5)^2 < 34 - \lambda \]

Calculating the left side:
\[ (2 - 3)^2 + (4 - 5)^2 = 1^2 + (-1)^2 = 1 + 1 = 2 \]

Thus, we have:
\[ 2 < 34 - \lambda \]
\[ \lambda < 34 - 2 \]
\[ \lambda < 32 \]

### Step 3: Condition for the circle not to cut the axes
For the circle not to cut the axes, the following conditions must hold:
1. \(g^2 < c\) for the x-axis (where \(c = \lambda\))
2. \(f^2 < c\) for the y-axis (where \(c = \lambda\))

From the circle equation:
- Coefficient of \(x\) is \(-6\), thus \(2g = -6 \Rightarrow g = -3\)
- Coefficient of \(y\) is \(-10\), thus \(2f = -10 \Rightarrow f = -5\)

Now, applying the conditions:
1. For the x-axis:
\[ g^2 < \lambda \Rightarrow (-3)^2 < \lambda \Rightarrow 9 < \lambda \]

2. For the y-axis:
\[ f^2 < \lambda \Rightarrow (-5)^2 < \lambda \Rightarrow 25 < \lambda \]

### Step 4: Combine the inequalities
Now we have the following inequalities:
1. \(9 < \lambda\)
2. \(25 < \lambda\)
3. \(\lambda < 32\)

The most restrictive conditions are:
\[ 25 < \lambda < 32 \]

### Final Answer
Thus, the range of \(\lambda\) is:
\[ \lambda \in (25, 32) \]