Number of circle touching both the axes and the line `x+y=4` is greater than or equal to : (a) 1 (b) 2 (c) 3 (d) 4

Options

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

Answer

To solve the problem of finding the number of circles that touch both the axes and the line \(x + y = 4\), we can follow these steps:

### Step 1: Understand the Geometry
We need to visualize the situation. The line \(x + y = 4\) intersects the x-axis at the point (4, 0) and the y-axis at the point (0, 4). This line divides the first quadrant into two regions.

**Hint:** Sketch the coordinate axes and the line \(x + y = 4\) to better understand the area in which the circles can be drawn.

### Step 2: Determine the Circle's Properties
A circle that touches both axes will have its center at a point \((r, r)\), where \(r\) is the radius of the circle. The equation of such a circle can be written as:
\[ (x - r)^2 + (y - r)^2 = r^2 \]

**Hint:** Remember that the distance from the center of the circle to the axes must equal the radius \(r\).

### Step 3: Find the Condition for Touching the Line
For the circle to touch the line \(x + y = 4\), the distance from the center \((r, r)\) to the line must equal the radius \(r\). The formula for the distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For the line \(x + y - 4 = 0\), we have \(A = 1\), \(B = 1\), and \(C = -4\).

**Hint:** Substitute the coordinates of the center \((r, r)\) into the distance formula.

### Step 4: Set Up the Equation
Substituting \((r, r)\) into the distance formula:
\[ d = \frac{|1 \cdot r + 1 \cdot r - 4|}{\sqrt{1^2 + 1^2}} = \frac{|2r - 4|}{\sqrt{2}} \]
Setting this equal to the radius \(r\):
\[ \frac{|2r - 4|}{\sqrt{2}} = r \]

**Hint:** Simplify this equation to find the values of \(r\).

### Step 5: Solve the Equation
Cross-multiplying gives:
\[ |2r - 4| = r\sqrt{2} \]
This leads to two cases:
1. \(2r - 4 = r\sqrt{2}\)
2. \(2r - 4 = -r\sqrt{2}\)

**Hint:** Solve both cases separately to find the possible values of \(r\).

### Step 6: Analyze the Solutions
After solving both cases, you will find that there are multiple values of \(r\) that satisfy the conditions. Each valid \(r\) corresponds to a circle that touches both axes and the line.

**Hint:** Count the number of valid solutions for \(r\) to determine how many circles can be drawn.

### Conclusion
Based on the analysis, we find that there are at least 4 circles that can satisfy the conditions given in the problem.

Thus, the answer to the question is:
**(d) 4**