A line touching the circle at a point can be called a **tangent to a circle**. At the tangency point, the radius is always perpendicular to the tangent of a circle.

The following are the properties of tangents:

- A tangent will never cross the circle but touches it.
- An angle is formed by the chord and tangent, which is the same as the angle that is being inscribed on the opposite side of the chord.
- The tangents of a circle are equal if it is emerging from the same external point.

## What Are Common Tangents in Coordinate Geometry?

A line which is tangent to more than one circle is called a **common tangent**. The tangents can be classified into common tangents that are internal and external. An internal tangent is a line segment, which passes through the centre of the two circles, whereas the external common tangents do not.

**Direct Common Tangents**

It is the tangents dividing externally in the ratio of the radii and meeting on the line of centres.

**Transverse Common Tangents**

It is the tangents dividing internally in the ratio of the radii and meeting on the line of centres.

## Common Tangents in Coordinate Geometry Condition and Formulae

**Number of Tangents Condition**

- 4 common tangents r
_{1}+ r_{2}< c_{1}c_{2} - 3 common tangents r
_{1}+ r_{2}= c_{1}c_{2} - 2 common tangents |r
_{1}– r_{2}| < c_{1}c_{2}< r_{1}+ r_{2} - 1 common tangent |r
_{1}– r_{2}| = c_{1}c_{2} - no common tangents c
_{1}c_{2}< |r_{1}– r_{2}|

The formula for the tangents of the two circles with respective centres and radii.

(i) Internally: If |C_{1} C_{2}| = |r_{2} – r_{1}| and the point of contact is ((r_{1} x_{2} – r_{2} x_{1})/(r_{1 }+ r_{2}) , (r_{1}y_{2} – r_{2}y_{1})/(r_{1} + r_{2})).

(ii) Externally: If |C_{1} C_{2}| = |r_{2} – r_{1}| and the point of contact is ((r_{1} x_{2} + r_{2} x_{1})/(r_{1 }+ r_{2}) , (r_{1}y_{2 }+ r_{2}y_{1})/(r_{1} + r_{2})).

Also, Read:

## Solved Problems on Common Tangents

**Example 1: **What is the angle between the two tangents from the origin to the circle (x − 7)^{2 }+ (y + 1)^{2 }= 25?

**Solution: **

Any line through (0, 0) be y − mx = 0, and it is a tangent to circle (x − 7)^{2 }+ (y + 1)^{2 }= 25, if [−1 − 7m] / √[1 + m^{2}] = 5

⇒ m = 3 / 4, −4 / 3

Therefore, the product of both slopes is -1

i.e., [3 / 4]* [−4 / 3] = −1

Hence, the angle between the two tangents is π/2.

**Example 2: **A pair of tangents are drawn from the origin to the circle x^{2 }+ y^{2} + 20 (x + y) + 20 = 0. Find the equation of the pair of tangents.

**Solution: **

The equation of pair of tangents is given by SS_{1} = T^{2}.

Here, S = x^{2 }+ y^{2} + 20 (x + y) + 20, S_{1 }= 20

T = 10 (x + y) + 20

∴ SS_{1} = T^{2}

⇒20 {x^{2 }+ y^{2} + 20 (x + y) + 20} = 10^{2} (x + y+ 2)^{2}

⇒4x^{2 }+ 4y^{2} + 10xy = 0

⇒2x^{2 }+ 2y^{2} + 5xy = 0

**Example 3: **What are the equations of the tangents to the circle x^{2} + y^{2} −6x + 4y = 12, which are parallel to the straight line 4x + 3y + 5 = 0?

**Solution: **

Let equation of tangent be 4x + 3y + 5 = 0, then √[9 + 4 + 12] =∣[4(3) + 3(−2) + k] / √[16+9]∣

⇒ 6 + k = ±25

⇒ k = 19 and −31

Hence, the tangents are 4x + 3y + 19 = 0 and 4x + 3y −31 = 0.

**Example 4: **Calculate the area of the triangle formed by the tangents from the points (h, k) to the circle x^{2} + y^{2} = a^{2} and the line joining their points of contact.

**Solution: **

Equation of chord of contact AB is xh + yk = a^{2} …..(i)

OM = Length of the perpendicular from O (0, 0) on line (i) = a^{2} / √[h^{2} + k^{2}]

∴ AB = 2AM = 2 √OA^{2 }− OM^{2}

= 2a √[h^{2} + k^{2}−a^{2}] / √[h^{2} + k^{2}]

Also, PM = Length of perpendicular from P(h, k) to the line (i) is [h^{2} + k^{2}−a^{2}] / √[h^{2} + k^{2}]. Therefore, the required area of triangle PAB = [1 / 2] * AB * PM = [a (h^{2} + k^{2}−a^{2})^{3/2}] / [h^{2} + k^{2}].

**Example 5: **Find the number of common tangents to the circles x^{2} + y^{2} − 4x − 6y − 12 = 0 and x^{2} + y^{2} + 6x + 18y + 26 = 0.

**Solution:**

Centres of circles are C_{1} (2, 3) and C_{2} (−3, −9), and their radii are r_{1} = 5 and r_{2} = 8. Obviously, r_{1 }+ r_{2} = C_{1}C_{2,} i.e., circles touch each other externally.

Hence, there are three common tangents.

**Example 6:** What is the gradient of the tangent line at the point (acosα, asinα) to the circle x^{2} + y^{2} = a^{2}?

**Solution:**

The equation of a tangent at (a cos α, a sin α) to the circle x^{2} + y^{2} = a^{2} is a x cos α + a y sin α = a^{2}.

Hence, its gradient is [−a cos α] / [a sin α] = − cot α.

**Example 7:** Find the number of common tangents to the circles x^{2} + y^{2} + 2x + 8y – 23 = 0 and x^{2} + y^{2} – 4x – 10y – 19 = 0 is

**Solution: **

Let

x^{2} + y^{2} + 2x + 8y – 23 = 0….(1)

And x^{2} + y^{2} – 4x – 10y – 19 = 0 …..(2)

The centre of the circle is given by C (-g, -f).

The centre of the circle (1) is (-1, -4), and of the circle (2) is (2, 5).

The radius of the circle is given by

The above two conditions are obtained.

There exist 3 scenarios.

- Two circles intersect each other.
- Two circles just touch the outer surface of each other.
- Two circles don’t touch each other.

This problem is two circles do not touch each other. Hence, the number of tangents is 2.

**Example 8:** If a > 2b > 0, then the positive value of m for which

^{2}+ y

^{2}= b

^{2}and (x – a)

^{2}+ y

^{2}= b

^{2}is

**Solution:**

Any tangent to x^{2} + y^{2} = b^{2} is

^{2}+ y

^{2}= b

^{2}, if

**Example 9:** The equation of tangent to the curve y = 2 cos x at x = π/4 is

**Solution:**

At x = π/4,

**Example 10:** The equation of the tangent to the curve (1 + x^{2})y = 2 – x, where it crosses the x-axis, is

**Solution:**

(1 + x^{2})y = 2 – x ……(i)

It meets x-axis, where y = 0 i.e., 0 = 2 – x ⇒ x = 2.

So, (i) meets the x-axis at the point (2, 0).

Also, from (i),

The slope of the tangent at (2, 0) is,

Equation of tangent at (2, 0) is

## Frequently Asked Questions

### What do you mean by a tangent of a circle?

The line that touches the circle only at one point is called the tangent to a circle.

### How many tangents can a circle have?

A circle can have an infinite number of tangents.

### What do you mean by a common tangent?

A common tangent is a line which is tangent to more than one circle.

### How many common tangents will there be for 2 circles intersecting each other at 2 points?

There will be two common tangents for 2 circles intersecting each other at 2 points.

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