Two circles with radii 6 cm and 2 cm have their centers 10 cm apart. How many common tangents can be drawn between them, and what type(s) of tangents are they?
- (a)4 tangents: 2 direct and 2 transverse
- (b)3 tangents: 2 direct and 1 transverse
- (c)2 tangents: only direct tangents
- (d)0 tangents: the larger circle completely contains the smaller one
Answer
Why
Correct — A. Compare the centre distance with the sum and the difference of the radii.
Sum of radii: 6 + 2 = 8 cm
Difference of radii: 6 − 2 = 4 cm
Centre distance: 10 cm > 8 cm
So the circles are separate, neither touching nor overlapping
Separate circles have 2 direct and 2 transverse common tangents
Total: 4 → option (a)
Why the others are wrong
- (b)3 tangents: 2 direct and 1 transverse — 3 tangents (2 direct, 1 transverse) needs external touching, centres exactly 6 + 2 = 8 cm apart. At 10 cm there is a gap, and the single transverse tangent becomes two.
- (c)2 tangents: only direct tangents — Direct tangents alone belong to intersecting circles, centres between 4 cm and 8 cm apart. At 10 cm the circles do not meet, so transverse tangents exist too.
- (d)0 tangents: the larger circle completely contains the smaller one — Zero common tangents means one circle sits inside the other, centres less than 6 − 2 = 4 cm apart. At 10 cm the smaller circle lies wholly outside the larger.
Concept
Count common tangents by comparing d with r₁ + r₂ and |r₁ − r₂|:
d > r₁ + r₂: separate, 4 (2 direct, 2 transverse)
d = r₁ + r₂: touching externally, 3
|r₁ − r₂| < d < r₁ + r₂: intersecting, 2 (both direct)
d = |r₁ − r₂|: touching internally, 1
d < |r₁ − r₂|: one inside the other, 0
Transverse tangents cross the line of centres between the circles, so they need a gap to pass through. When the circles touch externally the two merge into one tangent at the point of contact, which is why the count drops from 4 to 3.
Both lengths are real here: direct √(10² − 4²) = √84 ≈ 9.17 cm, transverse √(10² − 8²) = 6 cm.
Key facts
- Two separate circles (d > r₁ + r₂) have 4 common tangents: 2 direct and 2 transverse.
- Circles touching externally (d = r₁ + r₂) have 3 common tangents.
- Intersecting circles have 2 common tangents, both direct.
- Circles touching internally (d = r₁ − r₂) have 1 common tangent.
Study next
Common traps
- Comparing d with the difference of the radii (4 cm) instead of their sum (8 cm). Beating the difference rules out one circle inside the other, nothing more.
- Counting 4 tangents for circles that touch externally, where the two transverse tangents merge into one.
The same two circles, radii 6 cm and 2 cm with centres 10 cm apart, return at 24 Sep 2025, 12:30, Quant Q.21, which asks for the transverse tangent: √(10² − 8²) = 6 cm.
23 Sep 2025, 09:00, Quant Q.21 asks the centre distance for exactly three tangents (radii 5 cm and 3 cm give 8 cm).
15 Sep 2025, 12:30, Quant Q.22 asks the count for internal touching: 1.
Related PYQs
No directly related past PYQ was found.