SSC CGL 2024 Tier-I · 19 September 2024, Shift 2 · Quantitative Aptitude · Question 24 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.

PQ and RS are common tangents to two circles intersecting at A and B. A and B, when produced on both the sides, meet the tangents PQ and RS at X and Y, respectively. If AB = 3 cm and XY = 5 cm, then PQ is ________.

(a)6 cm
(b)4 cm
(c)5 cm
(d)3 cm

Answer

Why

Correct — B. X sits on line AB, the common chord produced, which is the radical axis of the two circles — every point on it has equal power with respect to both.

Power of X along the secant = XA × XB.
Power of X along the tangent = XP², and also XQ².
So XP = XQ: X is the midpoint of PQ.

The figure is symmetric about the line of centres, so XA = BY.
XA + AB + BY = XY gives 2 × XA + 3 = 5, so XA = 1 cm
XB = XA + AB = 1 + 3 = 4 cm

XP² = XA × XB = 1 × 4 = 4, so XP = 2 cm
PQ = XP + XQ = 2 × 2 = 4 cm → option (b)

Why the others are wrong

(a)
6 cm — 6 cm needs XP = 3 and so XA × XB = 9. With AB = 3 that forces XA ≈ 1.85 and XY ≈ 6.7 cm, not the 5 cm the stem states.
(c)
5 cm — 5 cm is XY copied across. XY runs along the common chord produced, PQ along the tangent — different segments, and PQ comes out shorter here.
(d)
3 cm — 3 cm is AB copied across. AB is the chord joining the two intersection points, while PQ is the tangent whose midpoint X lies 1 cm beyond A.

Concept

Every point on the radical axis of two circles has equal power with respect to both, and for two circles that intersect, the radical axis is the common chord produced.

A point X on it therefore satisfies XP² = XA × XB = XQ², so the tangent lengths from X to the two circles are equal. That one fact carries the question: X bisects PQ, and PQ is twice √(XA × XB).

No figure is printed with this stem, so sketch it. Two circles crossing at A and B, a tangent PQ touching them at P and Q, and the line AB extended cutting PQ at X and the far tangent RS at Y.

Key facts

The common chord of two intersecting circles, produced, bisects each common tangent.
Power of an external point X: XA × XB along a secant equals the square of the tangent length from X.
With AB = 3 and XY = 5, the 2 cm left over splits evenly, giving XA = BY = 1 cm.
XP = √(1 × 4) = 2 cm, so PQ = 4 cm.

Study next

Power of a point across secants, tangents and intersecting chordsLength of the direct common tangent, √(d² − (r₁ − r₂)²)Where the radical axis meets the line joining the two centres

Common traps

Assuming X is the midpoint of AB when it is the midpoint of PQ.
Taking XA as half of XY instead of half of XY − AB.
Answering 5 cm or 3 cm straight off the stem because no diagram is supplied.

Common tangents return on 11 Sep 2024, 16:00, Quant Q.16, where the direct common tangents to circles of radii 7 cm and 3 cm meet the line of centres produced, dividing it externally in the ratio 7 : 3.

Circle geometry also sits in this same paper at Quant Q.16, on a chord 6 cm from the centre of a circle of radius 10 cm, and at Quant Q.2 on a cyclic quadrilateral.

Related PYQs

No directly related past PYQ was found.