Common Tangents to Circles
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Direct answer
Four common tangents exist when two circles sit apart (d > r1 + r2): two direct, two transverse; external touching drops the count to three; overlapping (|r1 − r2| < d < r1 + r2) leaves only the two direct tangents; internal touching leaves one; and one circle inside the other (d < |r1 − r2|) leaves none. The lengths are one substitution each: direct tangent length = √(d² − (r1 − r2)²), transverse = √(d² − (r1 + r2)²), with d the distance between centres. Direct tangents meet at the external centre of similitude, dividing the line of centres in ratio r1 : r2 externally; transverse ones at the internal centre — fixed points from which tangent equations launch.
What you must remember
- The count ladder: d > r1 + r2 → 4 tangents; d = r1 + r2 → 3; |r1 − r2| < d < r1 + r2 → 2; d = |r1 − r2| → 1; d < |r1 − r2| → 0.
- Direct tangent length: √(d² − (r1 − r2)²) — radius difference inside the square root.
- Transverse tangent length: √(d² − (r1 + r2)²) — radius sum inside; exists only when the root is real, matching the count ladder.
- Centres of similitude: external centre divides centres in r1 : r2 externally, internal centre internally; all four common tangents pass through one or the other.
- Tangent-by-equation route: a line y = mx + c is a common tangent when the distance from each centre equals the matching radius — same sign of the two signed distances for direct tangents, opposite signs for transverse.
- Touching cases: at d = r1 + r2 the single transverse tangent is the common tangent at the touching point, perpendicular to the line of centres.
- Concentric special case: d = 0 with unequal radii gives no common tangents at all.
Counting before computing
Take the circles x² + y² = 4 and (x − 6)² + y² = 9. Centres (0, 0) and (6, 0), radii 2 and 3, so d = 6: the count ladder says 6 > 2 + 3 = 5, hence four common tangents. The direct tangent length is √(36 − (2 − 3)²) = √35 and the transverse length is √(36 − (2 + 3)²) = √11 — both irrational answers the options quote exactly. The centres of similitude follow immediately: the external centre divides (0, 0) and (6, 0) in 2 : 3 externally, landing at (−12, 0), while the internal centre divides internally at (12/5, 0). Every direct tangent passes through (−12, 0) and every transverse one through (12/5, 0), so a line through either point with slope m needs only one tangency condition to be fully determined. The professional sequence — centres, distance, count, lengths, similitude points — produces five exam quantities from about ninety seconds of arithmetic, which is why this configuration is worth rehearsing as a single unit.
Centres of similitude in play
JEE Main asks the count and the two lengths, with the standard distractors swapping the radius difference and sum between the two root formulas — √41 appearing where √11 belongs — and quoting 3 tangents for the internally touching case where the answer is 1. JEE Advanced asks for the equations: the line through the similitude centre, y = m(x + 12) in the worked example, then the distance from (0, 0) set equal to 2 gives |12m|/√(1 + m²) = 2, so 144m² = 4 + 4m², m² = 1/35 — the two direct tangents y = ±(x + 12)/√35, and the transverse pair follows the same route through (12/5, 0) with opposite-sign distances. The mixed breed — common tangent to a circle and a parabola, or to two conics — is the other Advanced face, solved by imposing both tangency conditions on one y = mx + c. The sign discipline is the whole game: direct tangents keep the circles on the same side (signed distances share a sign), transverse tangents sandwich them (opposite signs), and writing the two conditions with wrong signs manufactures tangents that touch nothing.
Frequently asked questions
How many common tangents do two separate circles have?
Four when d > r1 + r2: two direct (outer) and two transverse (crossing between them).
What is the length of a direct common tangent?
√(d² − (r1 − r2)²), with d the distance between centres — the radius difference sits inside the root.
Where do the direct common tangents intersect?
At the external centre of similitude, dividing the line of centres in the ratio r1 : r2 externally.
What happens to the tangent count when one circle is inside the other?
Zero common tangents when d < |r1 − r2|; at exact internal contact exactly one, at the touching point.
How do you find the equations of all common tangents?
Pass a line through the appropriate centre of similitude and impose the distance-from-centre equals radius condition — signs matching for direct, opposing for transverse.