Common Tangents to Two Circles

A circle’s tangent is a line that touches it at a point. The radius is always perpendicular to the tangent of a circle at the tangency point.

Tangents have the following qualities.

1. A tangent never crosses the circle, although it does touch it.

2. The chord and tangent form an angle, which is the same as the angle inscribed on the opposite side of the chord.

3. If a circle emerges from the same external point, its tangents are equal.

Tangents and Normals

A line that contacts the curve and has a slope that is suitable for the curve’s gradient/by-product could be the tangent at a degree on the curve. A line that intersects the curve and is perpendicular to the tangent can be normal at a degree on the curve. If its slope is provided by n, then the gradient/spinoff at that moment is the slope of the tangent.

Tangent to a circle is defined as a line that intersects the circle at one point. Within the circle, there will be just one tangent to some extent.

We’ll use the knowledge that the equation of a straight line passing through a point and having a gradient can be used to determine the equations of these lines.

Condition of Tangency

The tangent is only considered when it intersects a curve at a single point; otherwise, it is merely a line. Based on the point of tangency and its location in reference to the circle, we may establish the conditions for tangent as follows:

• When a point falls within the circle

• When a point falls within the circle

• When a point is outside the circle.

When a point falls inside the circle, no tangent can be drawn passing through it.

When a point falls on the circle, one tangent can be drawn passing through it.

From a point outside the circle, there are exactly two tangents to the circle.

Properties of Tangents

1. The tangent always comes into contact with the circle in a single spot.

2. It is perpendicular to the radius of the circle at the point of tangency.

3. It never has two intersecting points with the circle.

4. The length of tangents from an exterior point to a circle is the same.

Tangent Theories

1. The tangent to the circle is perpendicular to the radius of the circle at the point of contact.

2. The length of two tangents drawn from the circle’s external point is equal.

What are common tangents

A common tangent is a line that tangents multiple circles, generally two. Tangents can be divided into two types: internal and external tangents. Internal tangents are line segments that pass through the centre of the two circles whereas it is not the case for external common tangents.

Direct Common Tangents

Tangents that divide externally in the ratio of radii and meet on the line of centres.

Transverse Common Tangents

Internally, the tangents divide in the ratio of the radii and meet on the line of centres.

Conditions and Formulae for Common Tangents in Coordinate Geometry

Based on the relation between the distance between the centres of the two circles (C₁ and C₂ i.e., C₁C₂) and their radii r₁ and r₂, For the number of similar tangents between the two circles, we have five alternative scenarios. They are:

• 4 common tangents if we have r₁ + r₂ < C₁C₂

• 3 common tangents if we have r₁ + r₂ = C₁C₂

• 2 common tangents if we have  |r₁ – r₂| < C₁C₂ < r₁ + r₂ 

• 1 common tangent if we have |r₁ – r₂| = C₁C₂

• no common tangents if we have C₁C₂ < |r₁ – r₂|

Point of Contact

The point of contact is the point on a circle where a tangent is traced to a circle. It is also known as the point of tangency.

Formula to find the point of contact

The tangents of two circles with different centres C₁, C₂ and radii r₁, r₂ are calculated using this formula when the circles are touching each other

1.  Internally: If |C₁C₂| = |r₂ – r₁| then the point of contact is ((r₁ x₂ – r₂ x₁)/(r₁ + r₂) , (r₁y₂ – r₂y₁)/(r₁ + r₂)).

2. Externally: If |C₁C₂| = |r₂ – r₁| then the point of contact is ((r₁ x₂ + r₂ x₁)/(r₁ + r₂) , (r₁y₂ + r₂y₁)/(r₁ + r₂)).

Conclusion

A circle’s tangent is a line that touches it at a point. Within the circle, there will be just one tangent to some extent. A common tangent is a line that tangents multiple circles, generally two. Tangents can be divided into two types: internal and external tangents. Internal tangents are line segments that pass through the centre of the two circles whereas it is not the case for external common tangents.