SSC CGL 2025 Tier-I · 17 September 2025, Shift 2 · Quantitative Aptitude · Question 20 of 99

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 circle with radius x touches another circle with radius 2x externally. What is the length of a direct common tangent?

(a)2x
(b)3x
(c)2x√2
(d)3x√2

Answer

Why

Correct — C. Circles that touch externally have their centres r₁ + r₂ apart.

Centre distance: d = x + 2x = 3x
Direct tangent: L² = d² − (r₁ − r₂)²
Substitute: L² = (3x)² − (2x − x)² = 9x² − x² = 8x²
Take the root: L = √8 × x = 2x√2 → option (c)

Why the others are wrong

(a)
2x — 2x is too short: (2x)² = 4x², only half of L² = 9x² − x² = 8x². It is the correct 2x√2 with the √2 dropped.
(b)
3x — 3x is the distance between the centres, x + 2x. A direct tangent equals that distance only when the radii are equal; here they differ by x, so L² = 9x² − x².
(d)
3x√2 — 3x√2 ≈ 4.24x is longer than the 3x between the centres. A direct common tangent can never exceed the centre distance, because L² = d² − (r₁ − r₂)².

Concept

Direct common tangent: both circles lie on the same side of it. The radii to its touching points are both perpendicular to it, hence parallel.

A line through the smaller centre, parallel to the tangent, forms a right triangle: hypotenuse d, legs L and r₁ − r₂.

So L² = d² − (r₁ − r₂)². When the circles touch externally, d = r₁ + r₂, and this collapses to L = 2√(r₁r₂).

Shortcut check: 2√(x × 2x) = 2√(2x²) = 2x√2, the same answer without finding d.

A transverse common tangent crosses between the circles, and its leg is r₁ + r₂: M² = d² − (r₁ + r₂)². For circles that touch externally M = 0, because both touching points are the point of contact.

Key facts

Direct common tangent: L = √(d² − (r₁ − r₂)²).
Transverse common tangent: M = √(d² − (r₁ + r₂)²).
Circles touching externally: d = r₁ + r₂, so L = 2√(r₁r₂).

Study next

Transverse common tangent length for circles that do not meetFinding the centre distance from a given tangent lengthNumber of common tangents for each position of two circles

Common traps

Putting r₁ + r₂ into the direct-tangent formula, which gives 9x² − 9x² = 0.
Answering the centre distance 3x instead of the tangent length.

16 Sep 2025, 09:00, Quant Q.23 asks for the shortcut itself: circles touching externally have a direct common tangent T = 2√(r₁r₂).

15 Sep 2025, 09:00, Quant Q.22 runs the formula backwards: radii 12 cm and 4 cm with a 15 cm tangent give d² = 15² + 8² = 289, so d = 17 cm.

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.