SSC CGL 2024 Tier-I · 10 September 2024, Shift 3 · Quantitative Aptitude · Question 22 of 100

Text as printed on a candidate's response sheet for this shift; answer per SSC's tentative key as marked on that sheet. Solution last reviewed 6 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. The questions and options here are reproduced as printed on a candidate's own response sheet for this shift — the Commission's text, unchanged. Each answer is SSC's tentative key as marked on that sheet; SSC allowed challenges to it and never publishes a final key. The solutions are written by UnlockIAS.

The radii of two circle are 5 cm and 10 cm and the distance between their centres is 17 cm. Find the length of the transverse common tangent.

Answer

Why

Correct — D. A transverse (internal) common tangent crosses between the circles, so the two radii are added, not subtracted.

Length = √(d² − (r₁ + r₂)²)

r₁ + r₂ = 5 + 10 = 15
d = 17, so d² = 289 and 15² = 225

289 − 225 = 64
√64 = 8 cm → option (d).

The numbers are the 8–15–17 right triangle, so 8 can be written down without squaring anything.

Why the others are wrong

(a)
Fails the Pythagorean check: 6² + 15² = 36 + 225 = 261, and √261 ≈ 16.2, short of the 17 cm centre distance the stem gives.
(b)
7.5 is half of r₁ + r₂ = 15, not a tangent length. Squared it gives 56.25 + 225 = 281.25, a centre distance of about 16.8 cm rather than 17.
(c)
Fails the same check: 9² + 15² = 81 + 225 = 306, and √306 ≈ 17.5. Only 8 satisfies 8² + 15² = 289 = 17².

Concept

Two circles carry two families of common tangent, and the formulas differ by one sign.

Direct (external): √(d² − (r₁ − r₂)²) — it stays on one side of the line of centres.
Transverse (internal): √(d² − (r₁ + r₂)²) — it crosses between the circles.

Both are Pythagoras in disguise: shift the tangent onto the line of centres and the tangent length, the radius sum (or gap) and d form a right triangle.

A transverse common tangent only exists when the circles are fully apart — that is, d > r₁ + r₂.

Here 17 > 15, so it exists. Had the stem said 14 cm, the expression under the root would have gone negative and the question would have had no answer.

Key facts

Transverse common tangent length = √(d² − (r₁ + r₂)²).
Direct common tangent length = √(d² − (r₁ − r₂)²).
Here √(17² − 15²) = √64 = 8 cm.
A transverse common tangent exists only when the centre distance exceeds the sum of the radii.

Study next

How many common tangents two circles have in each relative positionLength of the tangent drawn from an external point to a circleWhere the two direct common tangents meet on the line of centres

Common traps

Using r₁ − r₂ = 5 and answering the direct tangent instead of the transverse one.
Mis-squaring 17 as 279 rather than 289 and losing the clean 64.
Reading transverse as the longer of the two tangents — it is always the shorter.

SSC tests the two tangent formulas as a pair, and runs them in both directions.

This very shift closes with the reverse at Quant Q.25, where both tangent lengths are given and the product of the radii has to come out.

Related PYQs

No directly related past PYQ was found.