Number of circles touching all the lines `x+4y+1=0, 2x+3y+3=0` and `x-6y+3=0` is

Options

A. 0
B. 2
C. 4
D. infinite

Answer

To solve the problem of finding the number of circles that touch all three lines given by the equations \(x + 4y + 1 = 0\), \(2x + 3y + 3 = 0\), and \(x - 6y + 3 = 0\), we will follow these steps:

### Step 1: Identify the lines
Let:
- Line \(L_1: x + 4y + 1 = 0\)
- Line \(L_2: 2x + 3y + 3 = 0\)
- Line \(L_3: x - 6y + 3 = 0\)

### Step 2: Find the intersection points of the lines
We will find the intersection points of the lines in pairs.

#### Intersection of \(L_1\) and \(L_2\)
1. From \(L_1\), express \(x\) in terms of \(y\):
\[ x = -4y - 1 \]
2. Substitute \(x\) into \(L_2\):
\[ 2(-4y - 1) + 3y + 3 = 0 \]
Simplifying:
\[ -8y - 2 + 3y + 3 = 0 \implies -5y + 1 = 0 \implies y = \frac{1}{5} \]
3. Substitute \(y\) back to find \(x\):
\[ x = -4\left(\frac{1}{5}\right) - 1 = -\frac{4}{5} - \frac{5}{5} = -\frac{9}{5} \]
4. Thus, the intersection point of \(L_1\) and \(L_2\) is:
\[ P_1 = \left(-\frac{9}{5}, \frac{1}{5}\right) \]

#### Intersection of \(L_1\) and \(L_3\)
1. From \(L_1\), we already have \(x = -4y - 1\).
2. From \(L_3\), express \(x\):
\[ x = 6y - 3 \]
3. Set the two expressions for \(x\) equal:
\[ -4y - 1 = 6y - 3 \]
Simplifying:
\[ -4y + 6y = -3 + 1 \implies 2y = -2 \implies y = \frac{1}{5} \]
4. Substitute \(y\) back to find \(x\):
\[ x = 6\left(\frac{1}{5}\right) - 3 = \frac{6}{5} - \frac{15}{5} = -\frac{9}{5} \]
5. Thus, the intersection point of \(L_1\) and \(L_3\) is also:
\[ P_2 = \left(-\frac{9}{5}, \frac{1}{5}\right) \]

#### Intersection of \(L_2\) and \(L_3\)
1. From \(L_2\), express \(x\):
\[ x = 6y - 3 \]
2. Substitute into \(L_2\):
\[ 2(6y - 3) + 3y + 3 = 0 \]
Simplifying:
\[ 12y - 6 + 3y + 3 = 0 \implies 15y - 3 = 0 \implies y = \frac{1}{5} \]
3. Substitute \(y\) back to find \(x\):
\[ x = 6\left(\frac{1}{5}\right) - 3 = -\frac{9}{5} \]
4. Thus, the intersection point of \(L_2\) and \(L_3\) is also:
\[ P_3 = \left(-\frac{9}{5}, \frac{1}{5}\right) \]

### Step 3: Analyze the intersection points
All three lines intersect at the same point \(P = \left(-\frac{9}{5}, \frac{1}{5}\right)\).

### Step 4: Determine the number of circles
Since all three lines intersect at a single point, there cannot be any circle that touches all three lines simultaneously. A circle can only touch a line at one point, and since all lines meet at one point, it is impossible to draw a circle that touches all three lines.

### Conclusion
The number of circles that can touch all three lines is:
\[ \text{Number of circles} = 0 \]