Find'the centre and radius of the circle
`(i) x^(2) +y^(2) + 4x - 1 = 0`
`(ii) 2 x^(2) + 2y^(2) = 3x - 5y + 7`

Answer

To find the center and radius of the given circles, we need to convert the given equations into the standard form of a circle, which is:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

where \((h, k)\) is the center and \(r\) is the radius.

### (i) For the equation: \(x^2 + y^2 + 4x - 1 = 0\)

**Step 1:** Rearrange the equation.

\[ x^2 + y^2 + 4x = 1 \]

**Step 2:** Complete the square for the \(x\) terms.

To complete the square for \(x^2 + 4x\), we take half of the coefficient of \(x\) (which is 4), square it, and add it to both sides:

\[ x^2 + 4x + 4 = 1 + 4 \]

This simplifies to:

\[ (x + 2)^2 + y^2 = 5 \]

**Step 3:** Write the equation in standard form.

Now we can express the equation as:

\[ (x + 2)^2 + (y - 0)^2 = 5 \]

**Step 4:** Identify the center and radius.

From the standard form \((x - h)^2 + (y - k)^2 = r^2\):

- Center \((h, k) = (-2, 0)\)
- Radius \(r = \sqrt{5}\)

### (ii) For the equation: \(2x^2 + 2y^2 = 3x - 5y + 7\)

**Step 1:** Divide the entire equation by 2 to simplify.

\[ x^2 + y^2 = \frac{3}{2}x - \frac{5}{2}y + \frac{7}{2} \]

**Step 2:** Rearrange the equation.

\[ x^2 - \frac{3}{2}x + y^2 + \frac{5}{2}y = \frac{7}{2} \]

**Step 3:** Complete the square for both \(x\) and \(y\).

For \(x^2 - \frac{3}{2}x\):

- Half of \(-\frac{3}{2}\) is \(-\frac{3}{4}\), and squaring it gives \(\left(-\frac{3}{4}\right)^2 = \frac{9}{16}\).

For \(y^2 + \frac{5}{2}y\):

- Half of \(\frac{5}{2}\) is \(\frac{5}{4}\), and squaring it gives \(\left(\frac{5}{4}\right)^2 = \frac{25}{16}\).

Now, we add these squares to both sides:

\[ x^2 - \frac{3}{2}x + \frac{9}{16} + y^2 + \frac{5}{2}y + \frac{25}{16} = \frac{7}{2} + \frac{9}{16} + \frac{25}{16} \]

**Step 4:** Simplify the left side.

The left side becomes:

\[ \left(x - \frac{3}{4}\right)^2 + \left(y + \frac{5}{4}\right)^2 \]

**Step 5:** Calculate the right side.

To combine the right side, we first convert \(\frac{7}{2}\) to sixteenths:

\[ \frac{7}{2} = \frac{56}{16} \]

Now adding:

\[ \frac{56}{16} + \frac{9}{16} + \frac{25}{16} = \frac{90}{16} \]

So we have:

\[ \left(x - \frac{3}{4}\right)^2 + \left(y + \frac{5}{4}\right)^2 = \frac{90}{16} \]

**Step 6:** Identify the center and radius.

From the standard form:

- Center \((h, k) = \left(\frac{3}{4}, -\frac{5}{4}\right)\)
- Radius \(r = \sqrt{\frac{90}{16}} = \frac{\sqrt{90}}{4}\)

### Summary of Results:
1. For the first equation:
- Center: \((-2, 0)\)
- Radius: \(\sqrt{5}\)

2. For the second equation:
- Center: \(\left(\frac{3}{4}, -\frac{5}{4}\right)\)
- Radius: \(\frac{\sqrt{90}}{4}\)