For two circles of radius 7 units and 7⁄2 units, whose centres are 15 units apart, what is the length of the direct common tangent in units? (Correct to 3 decimal places.)
- (a)14.586
- (b)18.654
- (c)16.584
- (d)15.486
Answer
Why
Correct — A. For two circles the direct common tangent has length L = √(d² − (r₁ − r₂)²).
r₁ − r₂ = 7 − 7⁄2 = 3.5
(r₁ − r₂)² = 12.25
d² = 15² = 225
225 − 12.25 = 212.75
L = √212.75 ≈ 14.586 units → option (a), correct to three decimal places.
Why the others are wrong
- (b)18.654 — 18.654 is longer than 15, the distance between the centres. The tangent is a leg of a right triangle whose hypotenuse is that 15, so it can never reach 15, let alone pass it.
- (c)16.584 — Square it and the claim collapses: 16.584² = 275.03, not the 212.75 the working requires. It is also longer than the 15-unit centre distance.
- (d)15.486 — 15.486 is the nearest miss, but it still exceeds 15, and 15.486² = 239.82 rather than 212.75.
Concept
Slide the smaller circle's centre along a parallel to the tangent until it meets the larger radius. The tangent then becomes one leg of a right triangle: the hypotenuse is d, the distance between the centres, and the other leg is the difference of the radii.
That gives L = √(d² − (r₁ − r₂)²). The transverse common tangent, the one that crosses between the circles, is built the same way but uses the sum of the radii: √(d² − (r₁ + r₂)²).
The 'correct to 3 decimal places' rider is a warning that 212.75 is not a perfect square.
All four options are the same five digits — 1, 4, 5, 6 and 8 — rearranged, so a half-remembered figure will not survive.
Key facts
- Direct (external) common tangent: √(d² − (r₁ − r₂)²).
- Transverse (internal) common tangent: √(d² − (r₁ + r₂)²), which needs d greater than r₁ + r₂.
- A tangent length can never exceed the distance between the centres, and equals it only when the two radii are equal.
- Here d = 15 while r₁ + r₂ = 10.5, so the circles lie wholly apart and both kinds of common tangent exist.
Study next
Common traps
- Using the sum of the radii instead of the difference, which gives the transverse tangent.
- Reading 7⁄2 as 7 and squaring a difference of 0.
- Skipping the sanity check that the answer must be under the 15-unit centre distance.
SSC moves the numbers around the same tangent geometry.
The direct common tangent, for two circles touching externally, at Quant Q.9 of 10 Sep 2024, 09:00 (radii 18 and 12).
The transverse common tangent, which uses the sum of the radii instead, at Quant Q.22 of 10 Sep 2024, 16:00 (radii 5 and 10, centres 17 apart).
Both formulas in one question at Quant Q.9 of 25 Sep 2024, 09:00, which asks for the sum of a direct and a transverse tangent (radii 15 and 9, centres 36 apart).
Related PYQs
No directly related past PYQ was found.