Types of Common Tangents Between Two Circles
Key Formulas
- 1. Equation of a Tangent to a Circle:
- 2. Distance Between Two Circle Centers:
- 3. Conditions for Common Tangents:
Steps to Find Common Tangents
Examples
- Example 1: Direct Common Tangents
- Example 2: All Tangents
Applications
FAQs

Common Tangents In Coordinate Geometry

Q: What number of normal digressions(Tangents) are there?

There can be three digressions in like manner. The one digression will be at the purpose of contacting where the two circles are contacting one another.

Q: What is the recipe for the length of normal digression?

The length of an immediate normal digression to two circles is √d2 -(r1 -r2 )2, where d is the distance between the focuses of the circles, and r1 and r2 are the radii of the given circles.

Types of Common Tangents Between Two Circles

Direct Common Tangents (DCT): The line touches both circles from the outside and does not intersect the line joining the centers.
Transverse Common Tangents (TCT): The line touches both circles, passing between them and intersecting the line joining the centers.

Key Formulas

1. Equation of a Tangent to a Circle:

For a circle with equation x² + y² + 2gx + 2fy + c = 0, the equation of a tangent in slope form is:

y = mx + sqrt(r²(1 + m²))

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where:

m is the slope of the tangent,
r is the radius of the circle.

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2. Distance Between Two Circle Centers:

If two circles have equations:

(x - x₁)² + (y - y₁)² = r₁² and (x - x₂)² + (y - y₂)² = r₂²,

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the distance between their centers is:

d = sqrt((x₂ - x₁)² + (y₂ - y₁)²)

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3. Conditions for Common Tangents:

Direct Common Tangents (DCT): Exists if d >= r₁ + r₂.
Transverse Common Tangents (TCT): Exists if d >= |r₁ - r₂|.
Number of Common Tangents:
- Four tangents: d > r₁ + r₂.
- Two tangents: d = r₁ + r₂ (touching circles externally).
- One tangent: d = |r₁ - r₂| (touching circles internally).
- No tangents: d < |r₁ - r₂| (one circle lies entirely inside the other).

Steps to Find Common Tangents

Write the general equation of a tangent to the first circle.
Solve for the condition that it also satisfies the equation of the second circle.
Simplify to get the equations of the common tangents.

Examples

Example 1: Direct Common Tangents

Find the equations of direct common tangents for circles:

(x - 1)² + y² = 4 and x² + y² = 1.

Centers: (1, 0) and (0, 0), Radii: 2 and 1.
Distance between centers: d = 1.
No direct common tangent exists as d < r₁ + r₂.

Example 2: All Tangents

For circles: (x - 3)² + y² = 4 and (x + 3)² + y² = 4.

Centers: (3, 0) and (-3, 0), Radii: 2.
Distance: d = 6.
There are four tangents (2 direct, 2 transverse).

Applications

Analyzing geometric properties of curves.
Solving optimization problems in calculus and physics.
Designing trajectories in robotics or animations.

FAQs

What number of normal digressions(Tangents) are there?

There can be three digressions in like manner. The one digression will be at the purpose of contacting where the two circles are contacting one another.

What is the recipe for the length of normal digression?

The length of an immediate normal digression to two circles is √d2 -(r1 -r2 )2, where d is the distance between the focuses of the circles, and r1 and r2 are the radii of the given circles.