To solve the problem, we need to analyze the given equation of the circle and determine the conditions under which its radius does not exceed 5. The equation given is:
\[ x^2 + y^2 + 7x + (1 - \lambda)y + 5 = 0 \]
### Step 1: Rewrite the equation in standard form
We can rearrange the equation to identify the center and radius of the circle. The standard form of a circle's equation is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius. We will complete the square for both \(x\) and \(y\).
1. For \(x\):
\[
x^2 + 7x = (x + \frac{7}{2})^2 - \frac{49}{4}
\]
2. For \(y\):
\[
y^2 + (1 - \lambda)y = (y + \frac{1 - \lambda}{2})^2 - \left(\frac{1 - \lambda}{2}\right)^2
\]
### Step 2: Substitute back into the equation
Substituting these completed squares back into the original equation gives:
\[
\left(x + \frac{7}{2}\right)^2 - \frac{49}{4} + \left(y + \frac{1 - \lambda}{2}\right)^2 - \left(\frac{1 - \lambda}{2}\right)^2 + 5 = 0
\]
### Step 3: Simplify the equation
Combining the constant terms:
\[
\left(x + \frac{7}{2}\right)^2 + \left(y + \frac{1 - \lambda}{2}\right)^2 = \frac{49}{4} + \left(\frac{1 - \lambda}{2}\right)^2 - 5
\]
### Step 4: Set up the radius condition
The radius \(r\) must satisfy the condition \(r^2 \leq 25\) (since the radius cannot exceed 5). Therefore:
\[
\frac{49}{4} + \left(\frac{1 - \lambda}{2}\right)^2 - 5 \leq 25
\]
### Step 5: Solve the inequality
Rearranging gives:
\[
\left(\frac{1 - \lambda}{2}\right)^2 \leq 25 - \frac{49}{4} + 5
\]
Calculating the right-hand side:
\[
25 - \frac{49}{4} + 5 = 25 + 5 - 12.25 = 17.75
\]
Thus, we have:
\[
\left(\frac{1 - \lambda}{2}\right)^2 \leq 17.75
\]
Taking the square root:
\[
-\sqrt{17.75} \leq \frac{1 - \lambda}{2} \leq \sqrt{17.75}
\]
### Step 6: Solve for \(\lambda\)
Multiplying through by 2 and rearranging gives:
\[
1 - 2\sqrt{17.75} \leq \lambda \leq 1 + 2\sqrt{17.75}
\]
Calculating \(\sqrt{17.75} \approx 4.22\):
\[
1 - 8.44 \leq \lambda \leq 1 + 8.44
\]
This simplifies to:
\[
-7.44 \leq \lambda \leq 9.44
\]
### Step 7: Count integral values of \(\lambda\)
The integral values of \(\lambda\) in this range are:
\[
\lambda = -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
\]
Counting these gives a total of 17 integral values.
### Final Answer
The number of integral values of \(\lambda\) for which the equation represents a circle whose radius cannot exceed 5 is **17**.
