Common Tangents to Two or More Circles
What is Common Tangents to Two or More Circles?
A line that is tangent to two or more circles at the same time.
Key formula / rule: Length of Direct Common Tangent
Key points
- Identify the types of common tangents.
- Determine the number of common tangents for given circle configurations.
- Calculate the length of direct and transverse common tangents.
- Apply the concept to solve geometry problems.
Common exam trap
Confusing the formulas for direct and transverse common tangent lengths.
Definitions
- Term
Common Tangent
- Meaning
A line that is tangent to two or more circles at the same time.
- Term
Direct Common Tangent
- Meaning
A common tangent where the two circles lie on the same side of the tangent line.
- Term
Transverse Common Tangent
- Meaning
A common tangent where the two circles lie on opposite sides of the tangent line.
Learning objectives
Identify the types of common tangents.
Determine the number of common tangents for given circle configurations.
Calculate the length of direct and transverse common tangents.
Apply the concept to solve geometry problems.
Formulae
- Name
Length of Direct Common Tangent
- Note
where d is the distance between centers and r1, r2 are the radii.
- Expression
Ldirect = \sqrt{d2 - (r1 - r2)2}
- Name
Length of Transverse Common Tangent
- Note
where d is the distance between centers and r1, r2 are the radii.
- Expression
Ltransverse = \sqrt{d2 - (r1 + r2)2}
Prerequisites
Basic geometry of circles (radius, diameter, center).
Understanding of tangents to a circle.
Pythagorean theorem.
Distance formula between two points.
Common mistakes
Confusing the formulas for direct and transverse common tangent lengths.
Incorrectly determining the number of common tangents based on circle positions.
Assuming all pairs of circles have 4 common tangents.
Calculation errors with square roots and differences/sums of radii.
Keywords
common tangent
direct tangent
transverse tangent
circle geometry
tangent length
relative position of circles
SSC CGL geometry
Practice preview
If two circles are completely separate from each other (disjoint), what is the maximum number of common tangents that can be drawn?…
easy
When two circles touch each other externally, how many common tangents can be drawn?…
easy
Two circles have radii 5 cm and 3 cm respectively. The distance between their centers is 10 cm. What is the length of the direct common tangent?…
medium