Equation of a circle which passes through (3,6) and touches the axes is

Options

A. `x^(2) + y^(2) + 6x + 6y + 3 = 0 `
B. ` x^(2) + y^(2) - 6x - 6y - 9 = 0 `
C. ` x^(2) + y^(2) - 6x - 6y + 9 = 0 `
D. none of these

Answer

To find the equation of a circle that passes through the point (3, 6) and touches both the axes, we can follow these steps:

### Step 1: Understand the properties of the circle
Since the circle touches both the axes, its center must be at a point (a, a) where 'a' is the radius of the circle. This is because the distance from the center to each axis must equal the radius.

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

### Step 3: Substitute the point (3, 6) into the equation
Since the circle passes through the point (3, 6), we can substitute x = 3 and y = 6 into the circle's equation:
\[ (3 - a)^2 + (6 - a)^2 = a^2 \]

### Step 4: Expand the equation
Now, let's expand the left-hand side:
\[ (3 - a)^2 = 9 - 6a + a^2 \]
\[ (6 - a)^2 = 36 - 12a + a^2 \]
Combining these, we have:
\[ 9 - 6a + a^2 + 36 - 12a + a^2 = a^2 \]
This simplifies to:
\[ 2a^2 - 18a + 45 = a^2 \]

### Step 5: Rearrange the equation
Now, we will rearrange the equation:
\[ 2a^2 - 18a + 45 - a^2 = 0 \]
This simplifies to:
\[ a^2 - 18a + 45 = 0 \]

### Step 6: Solve the quadratic equation
To solve the quadratic equation \( a^2 - 18a + 45 = 0 \), we can factor it:
\[ (a - 3)(a - 15) = 0 \]
Thus, we find:
\[ a = 3 \quad \text{or} \quad a = 15 \]

### Step 7: Write the equations of the circles
1. For \( a = 3 \):
\[ (x - 3)^2 + (y - 3)^2 = 3^2 \]
Expanding this gives:
\[ x^2 - 6x + 9 + y^2 - 6y + 9 = 9 \]
Simplifying, we get:
\[ x^2 + y^2 - 6x - 6y + 9 = 0 \]

2. For \( a = 15 \):
\[ (x - 15)^2 + (y - 15)^2 = 15^2 \]
Expanding this gives:
\[ x^2 - 30x + 225 + y^2 - 30y + 225 = 225 \]
Simplifying, we get:
\[ x^2 + y^2 - 30x - 30y + 225 = 0 \]

### Final Result
The equations of the circles that pass through (3, 6) and touch both axes are:
1. \( x^2 + y^2 - 6x - 6y + 9 = 0 \)
2. \( x^2 + y^2 - 30x - 30y + 225 = 0 \)