To solve the problem, we need to determine the set of values of \(\lambda\) for which the two circles are on opposite sides of the line given by the equation \(3x + 4y - \lambda = 0\).
### Step 1: Find the centers and radii of the circles.
1. **Circle 1**: The equation is \(x^2 + y^2 - 2x - 2y + 1 = 0\).
- Rearranging gives: \((x-1)^2 + (y-1)^2 = 1^2\).
- Center: \(C_1(1, 1)\), Radius: \(r_1 = 1\).
2. **Circle 2**: The equation is \(x^2 + y^2 - 18x - 2y + 78 = 0\).
- Rearranging gives: \((x-9)^2 + (y-1)^2 = 10^2\).
- Center: \(C_2(9, 1)\), Radius: \(r_2 = 10\).
### Step 2: Evaluate the line at the centers of the circles.
We substitute the coordinates of the centers into the line equation \(3x + 4y - \lambda = 0\).
1. For Circle 1 (Center \(C_1(1, 1)\)):
\[
3(1) + 4(1) - \lambda = 7 - \lambda
\]
2. For Circle 2 (Center \(C_2(9, 1)\)):
\[
3(9) + 4(1) - \lambda = 27 + 4 - \lambda = 31 - \lambda
\]
### Step 3: Set up the condition for opposite sides.
For the circles to be on opposite sides of the line, the product of the evaluations at the centers must be negative:
\[
(7 - \lambda)(31 - \lambda) < 0
\]
### Step 4: Solve the inequality.
1. Find the roots of the equation:
\[
7 - \lambda = 0 \implies \lambda = 7
\]
\[
31 - \lambda = 0 \implies \lambda = 31
\]
2. The critical points are \(\lambda = 7\) and \(\lambda = 31\). We analyze the intervals:
- For \(\lambda < 7\): Both terms are positive, product is positive.
- For \(7 < \lambda < 31\): One term is positive and the other is negative, product is negative.
- For \(\lambda > 31\): Both terms are negative, product is positive.
Thus, the solution for this inequality is:
\[
\lambda \in (7, 31)
\]
### Step 5: Check the distances from the line to the centers of the circles.
1. **Circle 1**: The distance from the center \(C_1(1, 1)\) to the line \(3x + 4y - \lambda = 0\) is given by:
\[
\text{Distance} = \frac{|3(1) + 4(1) - \lambda|}{\sqrt{3^2 + 4^2}} = \frac{|7 - \lambda|}{5}
\]
This distance must be greater than or equal to the radius \(1\):
\[
\frac{|7 - \lambda|}{5} \geq 1 \implies |7 - \lambda| \geq 5
\]
This gives us two cases:
- \(7 - \lambda \geq 5 \implies \lambda \leq 2\)
- \(\lambda - 7 \geq 5 \implies \lambda \geq 12\)
2. **Circle 2**: The distance from the center \(C_2(9, 1)\) to the line \(3x + 4y - \lambda = 0\) is:
\[
\text{Distance} = \frac{|31 - \lambda|}{5}
\]
This distance must be greater than or equal to the radius \(10\):
\[
\frac{|31 - \lambda|}{5} \geq 10 \implies |31 - \lambda| \geq 50
\]
This gives us two cases:
- \(31 - \lambda \geq 50 \implies \lambda \leq -19\)
- \(\lambda - 31 \geq 50 \implies \lambda \geq 81\)
### Step 6: Combine the results.
From the conditions derived:
- From Circle 1: \(\lambda \leq 2\) or \(\lambda \geq 12\)
- From Circle 2: \(\lambda \leq -19\) or \(\lambda \geq 81\)
The intersection of these conditions with the previous result \((7, 31)\) gives us:
\[
\lambda \in [12, 21]
\]
### Final Answer:
Thus, the set of all values of \(\lambda\) is the interval \([12, 21]\).
