Problem: Find the equation of a circle which passes through (2,-3) and (-4,5) and having centre on the line 4x + 3y + 1 = 0.

Solution:
To find the equation of the circle, we need to determine its centre and radius. We are given two points on the circle, so we can use the midpoint formula to find the centre.

Finding the Centre of the Circle:

The midpoint formula states that the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is:

((x1 + x2)/2, (y1 + y2)/2)

Using this formula, we can find the midpoint of the line segment that connects the given points (2,-3) and (-4,5). The midpoint is:

((2 + (-4))/2, (-3 + 5)/2) = (-1,1)

Therefore, the centre of the circle is (-1,1).

Finding the Radius of the Circle:

To find the radius of the circle, we need to find the distance between the centre and one of the given points. We can use the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Let's use the point (2,-3). The distance from the centre (-1,1) to (2,-3) is:

d = sqrt((-1 - 2)^2 + (1 - (-3))^2) = sqrt(9 + 16) = 5

Therefore, the radius of the circle is 5.

Equation of the Circle:

Now that we have the centre and radius of the circle, we can use the standard form of the equation of a circle:

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

where (h,k) is the centre of the circle and r is the radius. Substituting in our values, we get:

(x + 1)^2 + (y - 1)^2 = 25

However, we also know that the centre of the circle lies on the line 4x + 3y + 1 = 0. We can substitute y = (-4x - 1)/3 into the equation of the circle to eliminate y:

(x + 1)^2 + ((-4x - 1)/3 - 1)^2 = 25

Simplifying, we get:

(x + 1)^2 + (4/3 - 4x/3)^2 = 25

Multiplying both sides by 9, we get rid of the denominator:

9(x + 1)^2 + 16 - 32x + 9x^2 = 225

Expanding and simplifying, we get:

9x^2 - 54x + 192 = 0

Dividing by 3, we get:

3x^2 - 18x + 64 = 0

Using the quadratic formula, we can solve for x:

x = (18 +/- sqrt(18^2 - 4(3)(64)))/(2(3)) = 3 +/- 5i/3

Since the centre of the circle must be a real number