If two straight lines are coinciding with each other, then the number of points of intersection is/are:

Correct — A.

Coinciding lines are not two objects lying on top of each other. They are the same set of points — every point of one is a point of the other.

So their intersection is the whole line, and a line holds infinitely many points.

In coefficient form, a₁ ⁄ a₂ = b₁ ⁄ b₂ = c₁ ⁄ c₂ makes the second equation a multiple of the first, which is the infinitely many solutions case → option (a).

Why the others are wrong

• (b)
  unique point of intersection — A single common point is the case of two distinct, non-parallel lines, where a₁ ⁄ a₂ ≠ b₁ ⁄ b₂. Coinciding lines are not distinct, so they cannot meet at just one point.
• (c)
  finite number of point of intersection — A finite count above one is impossible for straight lines — two distinct lines meet at most once. Coinciding lines sit at the other extreme, sharing every one of their points.
• (d)
  no points of intersection — No common point describes parallel and distinct lines, where a₁ ⁄ a₂ = b₁ ⁄ b₂ ≠ c₁ ⁄ c₂. Coinciding lines share all their points, which is the opposite situation.

Two lines in a plane stand in exactly three relationships, and a pair of linear equations records which.

Intersecting, one common point: a₁ ⁄ a₂ ≠ b₁ ⁄ b₂, a unique solution.
Parallel and distinct, no common point: a₁ ⁄ a₂ = b₁ ⁄ b₂ ≠ c₁ ⁄ c₂, no solution.
Coincident, every point common: a₁ ⁄ a₂ = b₁ ⁄ b₂ = c₁ ⁄ c₂, infinitely many solutions.

Coincidence is the degenerate case: the second equation is the first multiplied by a constant and carries no new information.

Key facts

• Coincident lines share every point, so they have infinitely many points of intersection.
• For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, coincidence means a₁ ⁄ a₂ = b₁ ⁄ b₂ = c₁ ⁄ c₂.
• Two distinct straight lines in a plane meet in at most one point.

Common traps

• Reading coinciding as parallel and answering that there is no intersection.
• Answering one point because the drawing shows a single line.
• Forgetting that the c-ratio must match too, since a₁ ⁄ a₂ = b₁ ⁄ b₂ with a different c gives parallel distinct lines.

No directly related past PYQ was found.