The number of circles that touches all the three lines x+y-1=0, x-y-1=0 and y+1=0 is

Options

A. 2
B. 3
C. 4
D. 1

Answer

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

### Step 1: Identify the Lines
The equations of the lines are:
1. \(x + y - 1 = 0\) (Line 1)
2. \(x - y - 1 = 0\) (Line 2)
3. \(y + 1 = 0\) (Line 3)

### Step 2: Graph the Lines
To understand the configuration of these lines, we can graph them:
- **Line 1** intersects the axes at (1,0) and (0,1).
- **Line 2** intersects the axes at (1,0) and (0,-1).
- **Line 3** is a horizontal line at \(y = -1\).

Plotting these lines will show that they form a right-angled triangle.

### Step 3: Analyze the Triangle
The three lines form a triangle, and we need to determine how many circles can touch all three sides of this triangle.

### Step 4: Determine the Number of Circles
For any triangle, there are:
- 1 incircle (the circle that touches all three sides from the inside)
- 3 excircles (the circles that touch one side from the outside and the extensions of the other two sides)

Thus, the total number of circles that can be drawn touching all three lines is given by:
\[ \text{Total Circles} = \text{Number of Incircles} + \text{Number of Excircles} = 1 + 3 = 4 \]

### Final Answer
The number of circles that touch all three lines is **4**.

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