The equation of circle passing through (-1,-2) and touching both the axes is

Options

A. `x^(2)+y^(2)-2x-2y-1=0`
B. `x^(2)+y^(2)+2x+2y+1=0`
C. `x^(2)+y^(2)+10x+10y+25=0`
D. `x^(2)+y^(2)-2x-2y+1=0`

Answer

To find the equation of the circle that passes through the point (-1, -2) and touches both the axes, we can follow these steps:

### Step 1: Understand the Circle's Properties
Since the circle touches both the x-axis and y-axis, its center must be at a point (h, k) where both h and k are equal to the negative of the radius (r). Therefore, the center of the circle can be expressed as:
\[ (h, k) = (-r, -r) \]

### Step 2: Write the General Equation of the Circle
The general equation of a circle with center (h, k) and radius r is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting the center:
\[ (x + r)^2 + (y + r)^2 = r^2 \]

### Step 3: Substitute the Point (-1, -2)
Since the circle passes through the point (-1, -2), we substitute x = -1 and y = -2 into the equation:
\[ (-1 + r)^2 + (-2 + r)^2 = r^2 \]

### Step 4: Expand the Equation
Expanding the left-hand side:
\[ (-1 + r)^2 = (r - 1)^2 = r^2 - 2r + 1 \]
\[ (-2 + r)^2 = (r - 2)^2 = r^2 - 4r + 4 \]
Combining these:
\[ (r^2 - 2r + 1) + (r^2 - 4r + 4) = r^2 \]
This simplifies to:
\[ 2r^2 - 6r + 5 = r^2 \]

### Step 5: Rearrange the Equation
Rearranging gives:
\[ 2r^2 - 6r + 5 - r^2 = 0 \implies r^2 - 6r + 5 = 0 \]

### Step 6: Factor the Quadratic Equation
Factoring the quadratic:
\[ (r - 5)(r - 1) = 0 \]
Thus, we have two possible values for r:
\[ r = 5 \quad \text{or} \quad r = 1 \]

### Step 7: Find the Equations of the Circles
1. **For \( r = 5 \)**:
- Center: \( (-5, -5) \)
- Equation:
\[ (x + 5)^2 + (y + 5)^2 = 25 \]
Expanding:
\[ x^2 + 10x + 25 + y^2 + 10y + 25 = 25 \implies x^2 + y^2 + 10x + 10y + 25 = 0 \]

2. **For \( r = 1 \)**:
- Center: \( (-1, -1) \)
- Equation:
\[ (x + 1)^2 + (y + 1)^2 = 1 \]
Expanding:
\[ x^2 + 2x + 1 + y^2 + 2y + 1 = 1 \implies x^2 + y^2 + 2x + 2y + 1 = 0 \]

### Final Result
The equations of the circles are:
1. \( x^2 + y^2 + 10x + 10y + 25 = 0 \)
2. \( x^2 + y^2 + 2x + 2y + 1 = 0 \)