To find the equation of the circle that passes through the origin and cuts off intercepts of 3 and 4 from the axes, we can follow these steps:
### Step 1: Understand the intercepts
The circle cuts off intercepts of 3 on the x-axis and 4 on the y-axis. This means:
- The x-intercept is at the point (3, 0).
- The y-intercept is at the point (0, 4).
### Step 2: General equation of a circle
The general equation of a circle can be expressed as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
Since the circle passes through the origin (0, 0), we can substitute these coordinates into the equation.
### Step 3: Substitute the origin into the equation
Substituting (0, 0) into the equation gives:
\[ 0^2 + 0^2 + 2g(0) + 2f(0) + c = 0 \]
This simplifies to:
\[ c = 0 \]
Thus, the equation reduces to:
\[ x^2 + y^2 + 2gx + 2fy = 0 \]
### Step 4: Use the x-intercept
The x-intercept is given by the formula:
\[ 2\sqrt{g^2 - c} = 3 \]
Since \( c = 0 \), we have:
\[ 2\sqrt{g^2} = 3 \]
This simplifies to:
\[ 2|g| = 3 \]
Thus:
\[ |g| = \frac{3}{2} \]
This gives us two possible values for \( g \):
\[ g = \frac{3}{2} \quad \text{or} \quad g = -\frac{3}{2} \]
### Step 5: Use the y-intercept
The y-intercept is given by the formula:
\[ 2\sqrt{f^2 - c} = 4 \]
Again, since \( c = 0 \), we have:
\[ 2\sqrt{f^2} = 4 \]
This simplifies to:
\[ 2|f| = 4 \]
Thus:
\[ |f| = 2 \]
This gives us two possible values for \( f \):
\[ f = 2 \quad \text{or} \quad f = -2 \]
### Step 6: Combine the values of \( g \) and \( f \)
Now we can combine the values of \( g \) and \( f \):
1. \( g = \frac{3}{2}, f = 2 \)
2. \( g = \frac{3}{2}, f = -2 \)
3. \( g = -\frac{3}{2}, f = 2 \)
4. \( g = -\frac{3}{2}, f = -2 \)
### Step 7: Write the equations for each combination
Using the values of \( g \) and \( f \), we can write the equations of the circles:
1. For \( g = \frac{3}{2}, f = 2 \):
\[ x^2 + y^2 + 3x + 4y = 0 \]
2. For \( g = \frac{3}{2}, f = -2 \):
\[ x^2 + y^2 + 3x - 4y = 0 \]
3. For \( g = -\frac{3}{2}, f = 2 \):
\[ x^2 + y^2 - 3x + 4y = 0 \]
4. For \( g = -\frac{3}{2}, f = -2 \):
\[ x^2 + y^2 - 3x - 4y = 0 \]
### Final Answer
Thus, the equations of the circles that satisfy the given conditions are:
1. \( x^2 + y^2 + 3x + 4y = 0 \)
2. \( x^2 + y^2 + 3x - 4y = 0 \)
3. \( x^2 + y^2 - 3x + 4y = 0 \)
4. \( x^2 + y^2 - 3x - 4y = 0 \)
