SSC CGL 2024 Tier-I · 19 September 2024, Shift 3 · Quantitative Aptitude · Question 4 of 99

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.

From a point A, a tangent line is drawn to the circle of radius 7 units. From the same point, a secant is drawn to the circle which cuts the circle at B and C and the point B is near A than C. What is the length of BC in units if the length of the tangent to the circle from point A is 21 units and the length of AB is 14 units?

(a)18.5
(b)16.5
(c)15.5
(d)17.5

Answer

Why

Correct — D. From an external point the tangent and the secant are tied by AT² = AB × AC, where AT is the tangent length and B, C are where the secant meets the circle.

21² = 14 × AC
441 = 14 × AC
AC = 441⁄14 = 31.5

B lies between A and C, so BC is the leftover piece:
BC = AC − AB = 31.5 − 14 = 17.5 units → option (d).

Why the others are wrong

(a)
18.5 — 18.5 would put C at AC = 14 + 18.5 = 32.5, and then AB × AC = 14 × 32.5 = 455. The 21-unit tangent fixes that product at 441.
(b)
16.5 — 16.5 gives AC = 30.5 and a product of 14 × 30.5 = 427. The tangent is the geometric mean of AB and AC, so the product cannot drift below 441.
(c)
15.5 — 15.5 gives AC = 29.5 and a product of 413, well short of 441. Only AC = 31.5 satisfies AT² = AB × AC with AB = 14.

Concept

The power of a point says that every line drawn through an external point A meets the circle at two points whose distances from A have the same product.

For a secant that product is AB × AC. For a tangent the two meeting points coincide, so it is AT × AT. Equating them gives AT² = AB × AC — the tangent length is the geometric mean of the near and far secant segments, and the radius never enters the calculation.

The radius of 7 is not needed, and it does not sit comfortably with the rest of the numbers.

A 21-unit tangent puts A at √(21² + 7²) = √490 ≈ 22.1 units from the centre, so the nearest point of the circle would be about 15.1 units away — further than the stated AB = 14. Treat the radius as surplus data and answer on the tangent-secant relation the item is testing.

Key facts

Tangent-secant from an external point: AT² = AB × AC, with AC the whole secant.
Here 21² = 441 = 14 × 31.5, so AC = 31.5 and BC = 31.5 − 14 = 17.5.
The tangent length is the geometric mean of the near and the far secant segment.
BC is the difference AC − AB, never AC itself.

Study next

Two secants from one external point, where AB × AC = AD × AEIntersecting chords inside a circle, the same product rule with the point taken insideThe alternate segment theorem, which gives the angle between a tangent and a chord

Common traps

Reporting AC = 31.5 when the question asks for BC
Writing AT² = AB × BC, which drops the near segment from the whole secant
Hunting for a use for the radius, which plays no part in this calculation

The same tangent-and-secant figure returns as an angle question rather than a length one at 25 Sep 2024, 16:00, Quant Q.22, where a tangent TP and a secant TQR are drawn from T with ∠PTQ = 27° and ∠TPQ = 55°.

Related PYQs

No directly related past PYQ was found.