Numebr of circles touching all the lines `x+y-1=0, x-y=1=0` and `y+1=0` are

Options

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

Answer

To find the number of circles that can touch the 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 lines can be rewritten in slope-intercept form to understand their orientation:
1. \(x + y - 1 = 0\) can be rewritten as \(y = -x + 1\).
2. \(x - y - 1 = 0\) can be rewritten as \(y = x - 1\).
3. \(y + 1 = 0\) can be rewritten as \(y = -1\).

### Step 2: Graph the lines
By plotting these lines, we can see that they form a triangle. The lines intersect at three points, which are the vertices of the triangle.

### Step 3: Determine the type of triangle
The triangle formed by these lines is a right-angled triangle. This can be confirmed by checking the slopes of the lines:
- The slope of \(y = -x + 1\) is -1.
- The slope of \(y = x - 1\) is 1.
- The slope of \(y = -1\) is 0 (horizontal line).

Since the product of the slopes of the first two lines is -1, they are perpendicular, confirming that the triangle is right-angled.

### Step 4: Count the circles
Every triangle has:
- 1 incircle (the circle that touches all three sides from the inside).
- 3 excircles (the circles that touch one side and the extensions of the other two sides).

Thus, for our triangle:
- Number of circles = 1 (incircle) + 3 (excircles) = 4.

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

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