Equation of circle touching the lines `|x-2| + | y-3|=4` will be

Options

A. `(x-2)^(2)+ (y-3)^(2) =12`
B. `(x-2)^(2) + (y-3)^(2) =4`
C. `(x-2)^(2) + (y -3)^(2) =4`
D. `(x-2)^(2) + (y-3)^(2) =8`

Answer

To find the equation of a circle that touches the lines given by the equation \( |x-2| + |y-3| = 4 \), we can follow these steps:

### Step 1: Understand the equation of the lines
The equation \( |x-2| + |y-3| = 4 \) represents a diamond shape (or rhombus) centered at the point (2, 3) in the coordinate plane. This equation can be broken down into four cases based on the signs of \( x-2 \) and \( y-3 \).

### Step 2: Write the cases
1. **Case 1**: \( x - 2 \geq 0 \) and \( y - 3 \geq 0 \)
\[ x - 2 + y - 3 = 4 \implies x + y = 9 \]

2. **Case 2**: \( x - 2 \geq 0 \) and \( y - 3 < 0 \)
\[ x - 2 - (y - 3) = 4 \implies x - y = 3 \]

3. **Case 3**: \( x - 2 < 0 \) and \( y - 3 \geq 0 \)
\[ - (x - 2) + (y - 3) = 4 \implies -x + y = 5 \implies y = x + 5 \]

4. **Case 4**: \( x - 2 < 0 \) and \( y - 3 < 0 \)
\[ - (x - 2) - (y - 3) = 4 \implies -x - y = -1 \implies x + y = -1 \]

### Step 3: Identify the lines
The lines from the cases are:
1. \( x + y = 9 \)
2. \( x - y = 3 \)
3. \( y = x + 5 \)
4. \( x + y = -1 \)

### Step 4: Find the center of the circle
The center of the circle is at the point (2, 3), which is the point from which the distances to the lines will be calculated.

### Step 5: Calculate the radius
To find the radius of the circle, we need to calculate the perpendicular distance from the center (2, 3) to any of the lines. We can use the formula for the distance from a point to a line given by \( Ax + By + C = 0 \):
\[ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

Let's use the line \( x + y = 9 \) (or \( 1x + 1y - 9 = 0 \)):
- Here, \( A = 1, B = 1, C = -9 \)
- The point is \( (x_1, y_1) = (2, 3) \)

Calculating the distance:
\[ \text{Distance} = \frac{|1(2) + 1(3) - 9|}{\sqrt{1^2 + 1^2}} = \frac{|2 + 3 - 9|}{\sqrt{2}} = \frac{|-4|}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

### Step 6: Write the equation of the circle
The radius \( r = 2\sqrt{2} \), so \( r^2 = (2\sqrt{2})^2 = 8 \).

The standard equation of a circle with center \( (h, k) \) and radius \( r \) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = 2, k = 3, r^2 = 8 \):
\[ (x - 2)^2 + (y - 3)^2 = 8 \]

### Final Answer
The equation of the circle is:
\[ (x - 2)^2 + (y - 3)^2 = 8 \]

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