Given that we need to find the equation of the circle passing through the point (1,1) and having two diameters along the pair of lines x² - y² - 2x - 3 = 0.

We know that the point of intersection of the pair of straight lines ax² + 2hxy + by² + 2gx + 2fy + c = 0 is .

Let us assume the intersection point of the pair of lines be ‘O’.

⇒

⇒

⇒ O = (1,0)

We have a circle with centre (1,0) and passing through the point (1,1).

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.

⇒

⇒

⇒

⇒ r = 1

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 = 1

⇒ x² + y² - 2x = 0

∴The correct option is (d).