M and N are, respectively, the centres of two circles of radii 12 cm and 8 cm. QR is the common tangent to the circles. If MN = 16 cm, then what will be the length of QR?

Correct — D. MN = 16 cm is less than r₁ + r₂ = 20 cm, so the circles overlap and only the direct common tangent exists.

QR = √(MN² − (r₁ − r₂)²)
= √(16² − (12 − 8)²)
= √(256 − 16)
= √240 = 4√15 cm

That is the surd printed on option (d).

Why the others are wrong

• (a)
  Option (a) prints 5√17 cm, which is √425 ≈ 20.6 cm — longer than MN itself. Since QR = √(MN² − (r₁ − r₂)²), it can never exceed the 16 cm between the centres.
• (b)
  Option (b) prints 3√19 cm = √171. Landing there would need the radii to differ by √(256 − 171) = √85 ≈ 9.2 cm; they differ by 4 cm.
• (c)
  Option (c) prints 6√8 cm, that is 12√2 ≈ 16.97 cm, again longer than the 16 cm gap between the centres, so no tangent segment of that length can be drawn here.

For two circles whose centres are d apart with radii r₁ and r₂:

direct (external) common tangent = √(d² − (r₁ − r₂)²)
transverse (internal) common tangent = √(d² − (r₁ + r₂)²)

The transverse formula only means anything when d > r₁ + r₂, that is when the circles lie fully outside each other.

Here d = 16 and r₁ + r₂ = 20, so the circles cut each other and no transverse tangent exists. The direct formula needs only d > |r₁ − r₂|, and 16 > 4.

Key facts

• Direct common tangent length = √(d² − (r₁ − r₂)²), where d is the distance between the centres.
• Transverse common tangent length = √(d² − (r₁ + r₂)²), and it exists only when d > r₁ + r₂.
• Circles of radii 12 cm and 8 cm with centres 16 cm apart intersect, because 4 < 16 < 20.
• A direct common tangent is never longer than the distance between the two centres.

Common traps

• Using the transverse formula and meeting a negative number under the root without asking why
• Subtracting the radii but forgetting to square the difference before subtracting it
• Stopping at √240 when the options are printed as simplified surds

No directly related past PYQ was found.