SSC CGL 2025 Tier-I · 18 September 2025, Shift 1 · Quantitative Aptitude · Question 20 of 100

We transcribed this text from the pictures on a candidate's response sheet for this shift; the answer is the key SSC colour-marked on that sheet. Solution last reviewed 13 September 2026.

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A common internal tangent connecting two circles is 9 cm long. The distance between the circles' centers is 15 cm. If the larger circle has a radius of 8 cm, what is the radius of the smaller circle?

(a)4 cm
(b)1 cm
(c)6 cm
(d)7 cm

Answer

Why

Correct — A. For a common internal (transverse) tangent, length² = d² − (r₁ + r₂)², with the radii added.

9² = 15² − (8 + r)²
81 = 225 − (8 + r)²
(8 + r)² = 225 − 81 = 144
8 + r = 12, so r = 4 cm → option (a)

Check: 8 + 4 = 12, less than 15, so the circles are apart and an internal tangent exists.

Why the others are wrong

(b)
1 cm — r = 1 cm makes r₁ + r₂ = 9, and √(15² − 9²) = √144 = 12 cm, not 9 cm. The 9 is the tangent length, not the sum of the radii.
(c)
6 cm — r = 6 cm makes r₁ + r₂ = 14, and √(225 − 196) = √29 ≈ 5.4 cm, not 9 cm.
(d)
7 cm — r = 7 cm makes r₁ + r₂ = 15, equal to the distance between the centres. The circles would touch externally, and √(225 − 225) gives an internal tangent of length 0.

Concept

Two separate circles have two kinds of common tangent. A direct (external) tangent keeps both circles on one side: length = √(d² − (r₁ − r₂)²).

A transverse (internal) tangent passes between them and crosses the line of centres: length = √(d² − (r₁ + r₂)²).

Both come from one right triangle. Slide the tangent parallel to itself through a centre: one leg becomes r₁ ± r₂ and the hypotenuse is d.

The direct-tangent formula cannot produce an answer here. (8 − r)² = 144 gives r = −4 or r = 20, and neither is a smaller radius. That is the sign you picked the wrong formula.

Key facts

Direct common tangent: L = √(d² − (r₁ − r₂)²).
Transverse common tangent: L = √(d² − (r₁ + r₂)²).
Two transverse tangents need d > r₁ + r₂. At d = r₁ + r₂ the circles touch externally and those tangents merge into the tangent at the point of contact.

Study next

Direct common tangent length, with the radii subtractedNumber of common tangents for circles that are apart, touching or intersectingL² − M² = 4r₁r₂, linking the direct (L) and transverse (M) tangent lengths

Common traps

Using (r₁ − r₂)² from the direct-tangent formula. Here it gives r = 20 or −4, neither a smaller radius.
Answering 15 − 8 = 7 cm, which would make the circles touch and the internal tangent zero.

The transverse tangent is asked directly at 24 Sep 2025, 12:30, Quant Q.21 (radii 6 and 2 cm, centres 10 cm apart). The direct tangent, with the same 15 cm and an 8 cm radius, is 23 Sep 2025, 16:00, Quant Q.21.

At 10 Sep 2024, 16:00, Quant Q.25 the two formulas are combined to find the product of the radii.

Related PYQs

No directly related past PYQ was found.

Topic: Circle and its chords, tangents, angles subtended by chords of a circle, common tangents to two or more circles — all 140 SSC CGL questions on it, every solved shift.