LENGTH OF TANGENT TO A CIRCLE FROM AN EXTERNAL POINT

Using the formula given below, we find length of tangent drawn from the point (x₁, y₁).

Length of the tangent = √(x₁²+y₁²+2gx₁+2fy₁+c)

Example 1 :

Find the length of tangent to the circle

x²+y²-4x-3y+12  =  0

from the point (2,3)

Solution :

Length of the tangent = √(x₁²+y₁²+2gx₁+2fy₁+c)

Here x₁ = 2 and y₁ = 3

=  √2²+3²-4(2)-3(3)+12

=  √(4+9-8-9+12)

=  √(4+12-8)

=  √8

=  2√2 units       

Example 2 :

Show that the point (2,-1) lies out side the circle

x²+y²-6x-8y+12 = 0

Solution :

Here x₁  =  2 and y₁  =  -1

  =  √2²+(-1)²-6(2)-8(-1)+12

  =  √(4+1-16+8+12)

  =  √(5+8+12+1-8)

  =  √(26-8)

  =  √18

=  3√2 units 

The length of tangent is positive. So, the given point lies outside of the circle.

Example 3 :

Find the length of tangent to the circle  

x²+y²-4x+8y-5  =  0

from the point (2, 1) 

Solution :

Here x₁ = 2 and y₁ = 1

= √2²+1²-4(2)+8(1)-5

=  √(4+1-8+8-5)

=  0 

The length of tangent is zero. So, the given point lies on the circle.

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