Given that x - 4y + 7 = 0 is one of the diameters of the circle circumscribing the rectangle ABCD.

The coordinates of A and B are ( - 3,4) and (5,4).

Let us assume E be the mid - point of the line AB and ‘O’ be the centre of the circle.

⇒

⇒

⇒ - - - - - - (1)

Let us find the equation of the line AB.

We know that the equation of the line passing through the points (x₁, y₁) and (x₂, y₂) is given by

⇒

⇒

⇒ y = 4 ..... - (2)

From the figure, we can see that the line EO is perpendicular to the line AB.

So, the equation of the line EO is x = 1.

We find the centre as it is the point of intersection of the lines x - 4y + 7 = 0 and x = 1.

On solving this, we get the centre to be (1, 2).

We have a circle with centre (1, 2) and passing through the point (5,4).

We know that the radius of the circle is the distance between the centre and any point on the radius. So, we find the radius of the circle.

We know that the distance between the two points (x₁,y₁) and (x₂,y₂) is .

Let us assume r is the radius of the circle.

⇒

⇒

⇒

⇒ ..... (2)

We know that the equation of the circle with centre (p, q) and having radius ‘r’ is given by:

⇒ (x - p)² + (y - q)² = r²

Now we substitute the corresponding values in the equation:

⇒

⇒ x² - 2x + 1 + y² - 4y + 4 = 13

⇒ x² + y² - 2x - 4y - 8 = 0.

∴The equation of the circle is x² + y² - 2x - 4y - 8 = 0.