Two circles intersect at two points. Which of the following statements is true?

Correct — B. Count the common tangents from how the circles sit.

A transverse tangent keeps the two circles on opposite sides of it. Intersecting circles share points, so no transverse tangent exists.
The two direct tangents, with both circles on the same side, still exist.

Total: 2 common tangents → option (b)

Why the others are wrong

• (a)
  The internal tangent is longer than the external tangent — Intersecting circles have no internal (transverse) tangent to compare. An internal tangent keeps one circle on each side of it, which is impossible once the circles overlap.
• (c)
  The number of tangents depends on radii — The count is fixed by position, not by radii alone: it depends on the centre distance d against r₁ + r₂ and r₁ − r₂. Any two circles cutting at two points have 2.
• (d)
  They must be concentric — Concentric circles never cut at two points: unequal radii give no common point and equal radii give the same circle. Intersecting circles must have different centres.

Count common tangents from d, the distance between the centres, taking r₁ > r₂.

d > r₁ + r₂ (apart): 4, two direct and two transverse
d = r₁ + r₂ (touching outside): 3
r₁ − r₂ < d < r₁ + r₂ (intersecting): 2, both direct

d = r₁ − r₂ (touching inside): 1
d < r₁ − r₂ (one inside the other): 0

Key facts

• Two circles cutting at two points have exactly 2 common tangents, both direct.
• Circles touching externally have 3 common tangents, circles touching internally have 1.
• Circles lying apart (d > r₁ + r₂) have 4: two direct and two transverse.

Common traps

• Answering 4 by reflex — that count belongs to circles lying apart, not intersecting ones.
• Counting the common chord through the two intersection points as a tangent — it cuts both circles, so it is a secant.

No directly related past PYQ was found.