If `x^(2)+y^(2)+lambda x +(1-lambda)y+5=0` represents a circle whose radius cannot exceed 5, then the number of integral values of `lambda` is

Options

A. 14
B. 16
C. 18
D. none

Answer

To solve the problem, we need to analyze the given equation of the circle and determine the integral values of `lambda` such that the radius does not exceed 5.

### Step-by-Step Solution:

1. **Identify the General Form of the Circle Equation**:
The given equation is:
\[ x^2 + y^2 + \lambda x + (1 - \lambda)y + 5 = 0 \]
The general form of a circle is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
By comparing, we can identify:
- \( g = \frac{\lambda}{2} \)
- \( f = \frac{1 - \lambda}{2} \)
- \( c = 5 \)

2. **Calculate the Radius of the Circle**:
The formula for the radius \( R \) of the circle is given by:
\[ R = \sqrt{g^2 + f^2 - c} \]
Substituting the values of \( g \), \( f \), and \( c \):
\[ R = \sqrt{\left(\frac{\lambda}{2}\right)^2 + \left(\frac{1 - \lambda}{2}\right)^2 - 5} \]

3. **Simplify the Expression for Radius**:
Expanding the terms:
\[ R = \sqrt{\frac{\lambda^2}{4} + \frac{(1 - \lambda)^2}{4} - 5} \]
\[ = \sqrt{\frac{\lambda^2 + (1 - 2\lambda + \lambda^2)}{4} - 5} \]
\[ = \sqrt{\frac{2\lambda^2 - 2\lambda + 1}{4} - 5} \]
\[ = \sqrt{\frac{2\lambda^2 - 2\lambda + 1 - 20}{4}} \]
\[ = \sqrt{\frac{2\lambda^2 - 2\lambda - 19}{4}} \]

4. **Set the Condition for Radius**:
We know that the radius cannot exceed 5:
\[ R \leq 5 \]
Squaring both sides:
\[ \frac{2\lambda^2 - 2\lambda - 19}{4} \leq 25 \]
Multiplying through by 4:
\[ 2\lambda^2 - 2\lambda - 19 \leq 100 \]
Rearranging gives:
\[ 2\lambda^2 - 2\lambda - 119 \leq 0 \]

5. **Solve the Quadratic Inequality**:
First, we solve the equation:
\[ 2\lambda^2 - 2\lambda - 119 = 0 \]
Using the quadratic formula:
\[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 2 \cdot (-119)}}{2 \cdot 2} \]
\[ = \frac{2 \pm \sqrt{4 + 952}}{4} = \frac{2 \pm \sqrt{956}}{4} = \frac{2 \pm 2\sqrt{239}}{4} = \frac{1 \pm \sqrt{239}}{2} \]

6. **Approximate the Roots**:
Calculating \( \sqrt{239} \) gives approximately \( 15.46 \):
\[ \lambda_1 = \frac{1 - 15.46}{2} \approx -7.23 \quad \text{and} \quad \lambda_2 = \frac{1 + 15.46}{2} \approx 8.23 \]

7. **Determine the Integral Values**:
The inequality \( 2\lambda^2 - 2\lambda - 119 \leq 0 \) holds between the roots:
\[ -7.23 \leq \lambda \leq 8.23 \]
The integral values of \( \lambda \) in this range are:
\[ -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 \]
Counting these gives a total of 16 integral values.

### Final Answer:
The number of integral values of \( \lambda \) is **16**.