The equation of the circle passing through (0, 0) and making intercepts 2 units and 3 units on the x-axis and y-axis repectively, is

Options

A. `x^(2) + y^(2) - 6x - 2y = 0`
B. `x^(2) + y^(2) - 2x - 3y =0`
C. `x^(2) + Y^(2) -2x - 6y + 1 = 0`
D. `x^(2) + y^(2) -6x - 2y + 2 = 0`

Answer

To find the equation of the circle that passes through the origin (0, 0) and makes intercepts of 2 units on the x-axis and 3 units on the y-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Intercepts**:
- The circle makes intercepts of 2 units on the x-axis and 3 units on the y-axis.
- This means the circle intersects the x-axis at (2, 0) and (-2, 0) and the y-axis at (0, 3) and (0, -3).

2. **Determine the Center of the Circle**:
- The center of the circle will be at the midpoint of the intercepts on the axes.
- For the x-axis intercepts (2, 0) and (-2, 0), the midpoint is (0, 0).
- For the y-axis intercepts (0, 3) and (0, -3), the midpoint is (0, 0).
- However, since the circle passes through the origin (0, 0), we need to find a different center.

3. **Finding the Center**:
- The center of the circle can be determined by the intercepts.
- The x-coordinate of the center will be half of the x-intercept (which is 2), so it will be 1.
- The y-coordinate of the center will be half of the y-intercept (which is 3), so it will be 3/2.
- Therefore, the center of the circle is at (1, 3/2).

4. **Calculate the Radius**:
- The radius can be calculated using the distance formula from the center (1, 3/2) to the origin (0, 0).
- Using the distance formula:
\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(1 - 0)^2 + \left(\frac{3}{2} - 0\right)^2} \]
- This simplifies to:
\[ r = \sqrt{1^2 + \left(\frac{3}{2}\right)^2} = \sqrt{1 + \frac{9}{4}} = \sqrt{\frac{4}{4} + \frac{9}{4}} = \sqrt{\frac{13}{4}} = \frac{\sqrt{13}}{2} \]

5. **Write the Equation of the Circle**:
- The standard form of the equation of a circle is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
- Here, \(h = 1\), \(k = \frac{3}{2}\), and \(r^2 = \left(\frac{\sqrt{13}}{2}\right)^2 = \frac{13}{4}\).
- Substituting these values into the equation gives:
\[ (x - 1)^2 + \left(y - \frac{3}{2}\right)^2 = \frac{13}{4} \]

6. **Expand the Equation**:
- Expanding the left side:
\[ (x^2 - 2x + 1) + \left(y^2 - 3y + \frac{9}{4}\right) = \frac{13}{4} \]
- Combine and simplify:
\[ x^2 + y^2 - 2x - 3y + 1 + \frac{9}{4} = \frac{13}{4} \]
- Multiply through by 4 to eliminate the fraction:
\[ 4x^2 + 4y^2 - 8x - 12y + 4 + 9 = 13 \]
- This simplifies to:
\[ 4x^2 + 4y^2 - 8x - 12y = 0 \]
- Dividing by 4 gives:
\[ x^2 + y^2 - 2x - 3y = 0 \]

7. **Final Equation**:
- Thus, the equation of the circle is:
\[ x^2 + y^2 - 2x - 3y = 0 \]

### Conclusion:
The correct option is Option 2: \(x^2 + y^2 - 2x - 3y = 0\).