Find the lengths of the intercepts of the circle `3x^(2) + 3y^(2) - 5x + 3y = 0` on the co-ordinate axes

Answer

To find the lengths of the intercepts of the circle given by the equation \(3x^2 + 3y^2 - 5x + 3y = 0\) on the coordinate axes, we will follow these steps:

### Step 1: Rewrite the Circle Equation
First, we need to rewrite the equation in the standard form of a circle. The given equation is:
\[ 3x^2 + 3y^2 - 5x + 3y = 0 \]
We can divide the entire equation by 3 to simplify it:
\[ x^2 + y^2 - \frac{5}{3}x + y = 0 \]

### Step 2: Complete the Square
Next, we will complete the square for the \(x\) and \(y\) terms.

For \(x\):
\[ x^2 - \frac{5}{3}x \quad \text{can be completed as:} \quad \left(x - \frac{5}{6}\right)^2 - \left(\frac{5}{6}\right)^2 \]

For \(y\):
\[ y^2 + y \quad \text{can be completed as:} \quad \left(y + \frac{1}{2}\right)^2 - \left(\frac{1}{2}\right)^2 \]

Putting this together, we have:
\[ \left(x - \frac{5}{6}\right)^2 - \frac{25}{36} + \left(y + \frac{1}{2}\right)^2 - \frac{1}{4} = 0 \]

### Step 3: Simplify the Equation
Now, we can rearrange the equation:
\[ \left(x - \frac{5}{6}\right)^2 + \left(y + \frac{1}{2}\right)^2 = \frac{25}{36} + \frac{9}{36} \]
This simplifies to:
\[ \left(x - \frac{5}{6}\right)^2 + \left(y + \frac{1}{2}\right)^2 = \frac{34}{36} = \frac{17}{18} \]

### Step 4: Identify the Center and Radius
From the standard form \((x - h)^2 + (y - k)^2 = r^2\), we can identify:
- Center \((h, k) = \left(\frac{5}{6}, -\frac{1}{2}\right)\)
- Radius \(r = \sqrt{\frac{17}{18}}\)

### Step 5: Find the Perpendicular Distance to the Axes
To find the lengths of the intercepts on the axes, we need the perpendicular distances from the center to the axes.

1. **Distance to the x-axis (y = 0)**:
\[ d_x = \left| -\frac{1}{2} \right| = \frac{1}{2} \]

2. **Distance to the y-axis (x = 0)**:
\[ d_y = \left| \frac{5}{6} \right| = \frac{5}{6} \]

### Step 6: Calculate the Lengths of the Intercepts
Using the formula for the length of the intercepts:
\[ \text{Length of intercept} = 2\sqrt{r^2 - d^2} \]

1. **Length of x-intercept**:
\[ L_x = 2\sqrt{r^2 - d_x^2} = 2\sqrt{\frac{17}{18} - \left(\frac{1}{2}\right)^2} \]
\[ = 2\sqrt{\frac{17}{18} - \frac{1}{4}} = 2\sqrt{\frac{34}{36} - \frac{9}{36}} = 2\sqrt{\frac{25}{36}} = 2 \cdot \frac{5}{6} = \frac{5}{3} \]

2. **Length of y-intercept**:
\[ L_y = 2\sqrt{r^2 - d_y^2} = 2\sqrt{\frac{17}{18} - \left(\frac{5}{6}\right)^2} \]
\[ = 2\sqrt{\frac{17}{18} - \frac{25}{36}} = 2\sqrt{\frac{34}{36} - \frac{25}{36}} = 2\sqrt{\frac{9}{36}} = 2 \cdot \frac{3}{6} = 1 \]

### Final Result
The lengths of the intercepts of the circle on the coordinate axes are:
- Length of x-intercept: \(\frac{5}{3}\)
- Length of y-intercept: \(1\)