The point `(1,4)` are inside the circle `S: x^2+y^2-6x-10y+k=0`. What are the possible values of `k` if the circle S neither touches the axes nor cut them

Options

A. `p lt 29`
B. `p gt 25`
C. `p gt 27`
D. `p lt 27`

Answer

To find the possible values of \( k \) for the circle \( S: x^2 + y^2 - 6x - 10y + k = 0 \) such that the point \( (1, 4) \) is inside the circle and the circle neither touches nor cuts the axes, we will follow these steps:

### Step 1: Rewrite the Circle Equation
The equation of the circle can be rewritten in standard form. The general form is given as:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
To convert our equation, we complete the square for \( x \) and \( y \).

1. For \( x^2 - 6x \):
\[ x^2 - 6x = (x - 3)^2 - 9 \]

2. For \( y^2 - 10y \):
\[ y^2 - 10y = (y - 5)^2 - 25 \]

Thus, the equation becomes:
\[ (x - 3)^2 + (y - 5)^2 - 34 + k = 0 \implies (x - 3)^2 + (y - 5)^2 = 34 - k \]

### Step 2: Determine Conditions for the Circle
For the circle to neither touch nor cut the axes, we need to ensure that the distance from the center of the circle to the axes is greater than the radius.

- **Center of the Circle**: \( (3, 5) \)
- **Radius**: \( r = \sqrt{34 - k} \)

#### Condition 1: Circle does not touch the x-axis
The distance from the center to the x-axis is \( 5 \). Therefore:
\[ 5 > \sqrt{34 - k} \]
Squaring both sides:
\[ 25 > 34 - k \implies k > 9 \]

#### Condition 2: Circle does not touch the y-axis
The distance from the center to the y-axis is \( 3 \). Therefore:
\[ 3 > \sqrt{34 - k} \]
Squaring both sides:
\[ 9 > 34 - k \implies k > 25 \]

### Step 3: Point Inside the Circle Condition
The point \( (1, 4) \) must be inside the circle, which means:
\[ (1 - 3)^2 + (4 - 5)^2 < 34 - k \]
Calculating the left side:
\[ (-2)^2 + (-1)^2 = 4 + 1 = 5 < 34 - k \]
This implies:
\[ k < 29 \]

### Step 4: Combine All Conditions
From the conditions derived:
1. \( k > 25 \)
2. \( k < 29 \)

Thus, the possible values for \( k \) are:
\[ 25 < k < 29 \]

### Final Answer
The possible values of \( k \) are in the range:
\[ (25, 29) \]