If two circles are touching internally how many common tangents do they have?

Correct — B. The number of common tangents depends on where the circles sit.

Touching internally: the small circle lies inside the large one and meets it at one point, P.
Any other tangent to the small circle touches it at a point inside the large circle, so that line cuts the large circle twice.

At P, both circles' tangents are perpendicular to the line of centres, so they are the same line.
That leaves 1 common tangent → option (b)

Why the others are wrong

• (a)
  0 — 0 is for one circle inside the other without touching (distance between centres less than R − r). Here they touch at P, and the tangent there is shared.
• (c)
  2 — 2 common tangents belong to circles that cut at two points. As one slides inside until they touch internally, those two tangents merge into the one at the point of contact.
• (d)
  3 — 3 is the count for circles touching externally: two direct tangents plus the one at the point of contact. Internal contact keeps only that last one.

Compare the distance between the centres, d, with the sum R + r and the difference R − r of the radii.

Apart (d > R + r): 4 tangents. Touching externally (d = R + r): 3. Cutting at two points: 2.

Touching internally (d = R − r): 1. One inside the other, not touching (d < R − r): 0.

Key facts

• Two circles touching internally have exactly one common tangent, at the point of contact.
• For circles touching internally, the distance between the centres is R − r.
• Two circles touching externally have three common tangents.
• Two separate circles (d > R + r) have four common tangents: two direct and two transverse.

Common traps

• Carrying over 3, the count for circles touching externally.
• Answering 0 because one circle is inside the other, missing the shared tangent at the point of contact.

No directly related past PYQ was found.