SSC CGL 2024 Tier-I · 10 September 2024, Shift 1 · Quantitative Aptitude · Question 9 of 98

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 10 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.

Two circles of radii 18 cm and 12 cm touch each other externally. Find the length (in cm) of their direct common tangent.

Answer

Why

Correct — A. Touching externally fixes the centre distance at r₁ + r₂, which is the only extra fact the stem gives you.

d = 18 + 12 = 30 cm
Direct common tangent = √(d² − (r₁ − r₂)²)
= √(30² − 6²) = √(900 − 36) = √864
√864 = √(144 × 6) = 12√6 cm

The four choices are printed as images, and the one reading 12√6 is option (a).

Shortcut when the circles touch externally: length = 2√(r₁r₂) = 2√(18 × 12) = 2√216 = 12√6.

Why the others are wrong

(b)
Option (b) reads 18√6, which is just the larger radius carried into the answer. Squaring it gives 1,944 where the tangent length must square to 864.
(c)
Option (c) reads 15√6 ≈ 36.7 cm, longer than the 30 cm between the centres. The tangent is a leg of a right triangle whose hypotenuse is that 30 cm, so it can never exceed it.
(d)
Option (d) reads 10√6, which squares to 600. The required value is d² − (r₁ − r₂)² = 900 − 36 = 864, so this subtracts the wrong quantity.

Concept

Two circles touching externally have their centres exactly r₁ + r₂ apart. Touching internally puts them r₁ − r₂ apart.

For any two circles at centre distance d, the direct common tangent has length √(d² − (r₁ − r₂)²) and the transverse common tangent has length √(d² − (r₁ + r₂)²).

Putting d = r₁ + r₂ into the first gives √((r₁ + r₂)² − (r₁ − r₂)²) = √(4r₁r₂) = 2√(r₁r₂). So for externally touching circles the direct tangent is twice the geometric mean of the radii — here 2√216 = 12√6.

The transverse tangent has zero length in this configuration, because d = r₁ + r₂ makes d² − (r₁ + r₂)² = 0 and the circles meet at a single point. Only the direct tangent has a length to find.

Key facts

Circles touching externally have centre distance d = r₁ + r₂.
Direct common tangent length = √(d² − (r₁ − r₂)²).
Transverse common tangent length = √(d² − (r₁ + r₂)²).
For externally touching circles the direct common tangent equals 2√(r₁r₂).

Study next

Transverse common tangent length and the case where it vanishesCircles touching internally, where the centre distance is r₁ − r₂Where the direct common tangents cut the line joining the centres

Common traps

Using the transverse formula √(d² − (r₁ + r₂)²) and landing on zero.
Missing that d is not stated and must be built as 18 + 12 = 30.
Simplifying √864 as 4√54 or 6√24 and then finding no option that matches.

The centre distance is handed over outright at 09 Sep 2024, 09:00, Quant Q.17 (radii 22 and 10, d = 37) and at 10 Sep 2024, 16:00, Quant Q.22 (radii 5 and 10, d = 17, transverse tangent). Here the phrase touch each other externally is the clue you must convert into d = 30 yourself.

Related PYQs

No directly related past PYQ was found.