The distance between the centres of two circles of radii 22 cm and 10 cm is 37 cm. If the points of contact of a direct common tangent to these circles are M and Q, then find the length of the line segment MQ.

Correct — C. M and Q are the contact points of a direct (external) common tangent, whose length is √(d² − (r₁ − r₂)²).

Difference of radii: 22 − 10 = 12 cm
Square the centre distance: 37² = 1369
Square the difference: 12² = 144

Subtract: 1369 − 144 = 1225
MQ = √1225 = 35 cm → option (c).

Why the others are wrong

• (a)
  39 cm — 39 cm is what adding gives: √(37² + 12²) = √1513 ≈ 38.9. The formula subtracts the squared radius difference, it does not add it.
• (b)
  29 cm — 29 cm fits no reading of the formula — 29² = 841 would need the radii to differ by √(1369 − 841) ≈ 23 cm, not the 12 cm the stem gives.
• (d)
  25 cm — 25 cm is 37 − 12, subtracting the lengths themselves instead of their squares. The tangent, the centre line and the radius difference form a right triangle, so Pythagoras is compulsory.

Drop a perpendicular from the smaller centre onto the radius of the larger one at its tangent point. That builds a right triangle.

Its hypotenuse is the centre distance d, one leg is the tangent length, and the other leg is r₁ − r₂ for a direct tangent.

Pythagoras then gives length = √(d² − (r₁ − r₂)²). For a transverse tangent, the circles lie on opposite sides of the tangent line and the leg becomes r₁ + r₂ instead.

Key facts

• Direct (external) common tangent length = √(d² − (r₁ − r₂)²).
• Transverse (internal) common tangent length = √(d² − (r₁ + r₂)²), which needs d > r₁ + r₂.
• 12, 35, 37 is a Pythagorean triple, which is why 1369 − 144 = 1225 lands on a whole 35.

Common traps

• Using r₁ + r₂ = 32, which answers for the transverse tangent instead of the direct one
• Subtracting the lengths as 37 − 12 = 25 rather than subtracting their squares
• Taking d as the distance between the tangent points instead of between the centres

No directly related past PYQ was found.