Find the points on the curve x² + y² – 2x – 3 = 0 at which the tangents are parallel to the x-axis.

It is given that x² + y² – 2x – 3 = 0

Now, differentiating both sides with respect to x, we get

We know that the tangents are parallel to the x –axis if the slope of the tangent is 0 ie,

⇒ 1-x = 0

⇒ x = 1

But, x² + y² – 2x – 3 = 0 for x = 1

⇒ y² = 4

⇒ y = 2

Therefore, the points at which the tangents are parallel to the x-axis are (1,2) and (1, -2).