## About "How to prove if a line is tangent to a circle"

Here we are going to see how to prove if a line is tangent to a circle.

In order to prove the given line is a tangent to the circle, it has to satisfy the condition given below.

c2  = a2 (1 + m2)

Let us look into some example problems based on the above concept.

Example 1 :

If y = 3x + c is a tangent to the circle x+ y= 9, find the value of c.

Solution :

The condition for the line y = mx + c to be a tangent to

x2 + y2 = a2 is c = ± a √(1 + m2)

Here a = 3, m = 3

Applying the values of "a" and "m",  we get

c  =  ± 3 √(1 + 32)

c  =  ± 3 10

Hence the value of c is ± 3 10.

Example 2 :

Find the value of p if the line 3x + 4y − p = 0 is a tangent to the circle x2 + y2 = 16.

Solution :

The condition for the tangency is c2 = a2 (1 + m2) .

Here a2 = 16, m = −3/4, c =  p/4

c2  =  a2 (1 + m2)

p2/16  =  16 (1 + 9/16)

p2/16  =  16 (25/16)

p2/16  =  25

p2  =  25(16)

p = ± 20

Example 3 :

Find the value of p so that the line 3x + 4y − p = 0 is a tangent to x2 + y2 − 64 = 0

Solution :

Equation of the line 3x + 4y − p = 0

Equation of the circle x2 + y2  =  64

The condition for the tangency is c2 = a2 (1 + m2) .

Here a2 = 64, m = −3/4, c =  p/4

c2  =  a2 (1 + m2)

p2/16  =  64 (1 + (-3/4)2)

p2/16  =  64 (1 + 9/16)

p2/16  =  64 (25/16)

p2/16  =  4(25)

p2  =  4 ⋅ 25 ⋅ 16

p = ± 2 ⋅ 5 ⋅ 4

p = ± 40

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