To solve the given problems step by step, we will use the standard equation of a circle, which is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
### (i) Find the equation of the circle passing through the origin and having the center at (-4, -3).
**Step 1**: Identify the center and the point through which the circle passes.
- Center: \((-4, -3)\)
- Point: \((0, 0)\) (the origin)
**Step 2**: Write the general equation of the circle.
\[
(x + 4)^2 + (y + 3)^2 = r^2
\]
**Step 3**: Substitute the point (0, 0) into the equation to find \(r^2\).
\[
(0 + 4)^2 + (0 + 3)^2 = r^2
\]
\[
4^2 + 3^2 = r^2
\]
\[
16 + 9 = r^2
\]
\[
r^2 = 25
\]
**Step 4**: Write the final equation of the circle.
\[
(x + 4)^2 + (y + 3)^2 = 25
\]
### (ii) Find the equation of the circle passing through (-2, 3) and having center at (0, 0).
**Step 1**: Identify the center and the point through which the circle passes.
- Center: \((0, 0)\)
- Point: \((-2, 3)\)
**Step 2**: Write the general equation of the circle.
\[
x^2 + y^2 = r^2
\]
**Step 3**: Substitute the point (-2, 3) into the equation to find \(r^2\).
\[
(-2)^2 + (3)^2 = r^2
\]
\[
4 + 9 = r^2
\]
\[
r^2 = 13
\]
**Step 4**: Write the final equation of the circle.
\[
x^2 + y^2 = 13
\]
### (iii) Find the equation of the circle passing through (3, 4) and having the center at (-3, 4).
**Step 1**: Identify the center and the point through which the circle passes.
- Center: \((-3, 4)\)
- Point: \((3, 4)\)
**Step 2**: Write the general equation of the circle.
\[
(x + 3)^2 + (y - 4)^2 = r^2
\]
**Step 3**: Substitute the point (3, 4) into the equation to find \(r^2\).
\[
(3 + 3)^2 + (4 - 4)^2 = r^2
\]
\[
6^2 + 0^2 = r^2
\]
\[
36 = r^2
\]
**Step 4**: Write the final equation of the circle.
\[
(x + 3)^2 + (y - 4)^2 = 36
\]
### (iv) Find the equation of the circle whose center is (-1, 2) and which passes through (5, 6).
**Step 1**: Identify the center and the point through which the circle passes.
- Center: \((-1, 2)\)
- Point: \((5, 6)\)
**Step 2**: Write the general equation of the circle.
\[
(x + 1)^2 + (y - 2)^2 = r^2
\]
**Step 3**: Substitute the point (5, 6) into the equation to find \(r^2\).
\[
(5 + 1)^2 + (6 - 2)^2 = r^2
\]
\[
6^2 + 4^2 = r^2
\]
\[
36 + 16 = r^2
\]
\[
r^2 = 52
\]
**Step 4**: Write the final equation of the circle.
\[
(x + 1)^2 + (y - 2)^2 = 52
\]
