The positive integral value of `lambda`, for which line `4x + 3y - 16lambda = 0` lies between the circles `x^(2) + y^(2) - 4x - 4y + 4 = 0` and `x^(2) + y^(2) - 20x - 2y + 100 = 0`, and does not intersect either of the circles, may be

Options

A. (a)27
B. (b)30
C. (c)33
D. (d)36

Answer

To solve the problem, we need to find the positive integral value of \(\lambda\) for which the line \(4x + 3y - 16\lambda = 0\) lies between the two circles given by the equations \(x^2 + y^2 - 4x - 4y + 4 = 0\) and \(x^2 + y^2 - 20x - 2y + 100 = 0\), without intersecting either of the circles.

### Step 1: Rewrite the equations of the circles

We start by rewriting the equations of the circles in standard form.

1. **Circle 1:**
\[ x^2 + y^2 - 4x - 4y + 4 = 0 \]
Completing the square:
\[ (x^2 - 4x) + (y^2 - 4y) + 4 = 0 \implies (x-2)^2 + (y-2)^2 = 0 \]
This circle has center \(C_1(2, 2)\) and radius \(r_1 = 2\).

2. **Circle 2:**
\[ x^2 + y^2 - 20x - 2y + 100 = 0 \]
Completing the square:
\[ (x^2 - 20x) + (y^2 - 2y) + 100 = 0 \implies (x-10)^2 + (y-1)^2 = 1 \]
This circle has center \(C_2(10, 1)\) and radius \(r_2 = 10\).

### Step 2: Find the distance from the center of each circle to the line

The line equation is \(4x + 3y - 16\lambda = 0\). The distance \(D\) from a point \((x_0, y_0)\) to the line \(Ax + By + C = 0\) is given by:
\[ D = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

Using this formula, we calculate the distances from the centers of the circles to the line.

1. **Distance from \(C_1(2, 2)\):**
\[ D_1 = \frac{|4(2) + 3(2) - 16\lambda|}{\sqrt{4^2 + 3^2}} = \frac{|8 + 6 - 16\lambda|}{5} = \frac{|14 - 16\lambda|}{5} \]

2. **Distance from \(C_2(10, 1)\):**
\[ D_2 = \frac{|4(10) + 3(1) - 16\lambda|}{\sqrt{4^2 + 3^2}} = \frac{|40 + 3 - 16\lambda|}{5} = \frac{|43 - 16\lambda|}{5} \]

### Step 3: Set up inequalities for the distances

For the line to lie between the circles without intersecting them, the distance from the line to each center must be greater than the radius of the respective circle.

1. **For Circle 1:**
\[ \frac{|14 - 16\lambda|}{5} > 2 \implies |14 - 16\lambda| > 10 \]
This gives us two inequalities:
\[ 14 - 16\lambda > 10 \quad \text{or} \quad 14 - 16\lambda < -10 \]
Solving these:
- \(14 - 16\lambda > 10 \implies -16\lambda > -4 \implies \lambda < \frac{1}{4}\)
- \(14 - 16\lambda < -10 \implies -16\lambda < -24 \implies \lambda > \frac{3}{2}\)

2. **For Circle 2:**
\[ \frac{|43 - 16\lambda|}{5} > 10 \implies |43 - 16\lambda| > 50 \]
This gives us two inequalities:
\[ 43 - 16\lambda > 50 \quad \text{or} \quad 43 - 16\lambda < -50 \]
Solving these:
- \(43 - 16\lambda > 50 \implies -16\lambda > 7 \implies \lambda < -\frac{7}{16}\) (not applicable since \(\lambda\) is positive)
- \(43 - 16\lambda < -50 \implies -16\lambda < -93 \implies \lambda > \frac{93}{16} \approx 5.8125\)

### Step 4: Combine the inequalities

From Circle 1, we have:
- \(\lambda < \frac{1}{4}\) or \(\lambda > \frac{3}{2}\)

From Circle 2, we have:
- \(\lambda > \frac{93}{16}\)

Thus, the valid range for \(\lambda\) is:
\[ \lambda > \frac{93}{16} \approx 5.8125 \]

### Step 5: Find the positive integral values of \(\lambda\)

The smallest positive integer greater than \(5.8125\) is \(6\).

### Final Answer
The positive integral value of \(\lambda\) is \(6\).