The number of integral values of `lambda` for which the equation `x^(2)+y^(2)-2lambdax+2lambday+14=0` represents a circle whose radius cannot exceed 6, is

Options

A. 10
B. 11
C. 12
D. 9

Explanation

To find the number of integral values of \( \lambda \) for which the equation

\[ x^2 + y^2 - 2\lambda x + 2\lambda y + 14 = 0 \]

represents a circle whose radius cannot exceed 6, we can follow these steps:

### Step 1: Rewrite the equation in standard form
The given equation can be compared to the general form of a circle:

\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

From the equation, we identify:
- \( g = -\lambda \)
- \( f = \lambda \)
- \( c = 14 \)

### Step 2: Determine the radius of the circle
The radius \( r \) of the circle can be found using the formula:

\[ r = \sqrt{g^2 + f^2 - c} \]

Substituting the values of \( g \), \( f \), and \( c \):

\[ r = \sqrt{(-\lambda)^2 + (\lambda)^2 - 14} = \sqrt{\lambda^2 + \lambda^2 - 14} = \sqrt{2\lambda^2 - 14} \]

### Step 3: Set up the inequality for the radius
We need the radius to not exceed 6, so we set up the inequality:

\[ \sqrt{2\lambda^2 - 14} \leq 6 \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:

\[ 2\lambda^2 - 14 \leq 36 \]

### Step 5: Simplify the inequality
Rearranging the inequality:

\[ 2\lambda^2 \leq 50 \]

Dividing by 2:

\[ \lambda^2 \leq 25 \]

### Step 6: Determine the range for \( \lambda \)
Taking the square root of both sides, we find:

\[ -\sqrt{25} \leq \lambda \leq \sqrt{25} \]

This simplifies to:

\[ -5 \leq \lambda \leq 5 \]

### Step 7: Count the integral values of \( \lambda \)
The integral values of \( \lambda \) within this range are:

\[ -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 \]

Counting these values gives us a total of 11 integral values.

### Final Answer
Thus, the number of integral values of \( \lambda \) for which the equation represents a circle whose radius cannot exceed 6 is:

\[ \boxed{11} \]